1 Introduction

From a fundamental point of view, acceleration is one of the most important quantities in mechanics. In particular, when considering fluid mechanics, it represents the sum of both the viscous and the pressure forces involved within the flow. The relationship between these two quantities and the temporal derivative of the velocity is directly given by the Navier-Stokes equation:

$$\frac{Du_i}{Dt} = \frac{\partial u_i}{\partial t} + u_j\frac{\partial u_i}{\partial x_j} = - \frac{1}{\rho}\frac{\partial p}{\partial x_i} + \nu \frac{\partial^2 u_i}{\partial x_i \partial x_j} $$
(1)

where Du i /Dt is the Lagrangian acceleration, also called fluid-particle acceleration, ∂u i /∂t the local acceleration and u j ∂u i /∂x j the convective acceleration. From a practical point of view, it may be easier to use velocity fields measured around a body for the prediction of the pressure distribution and overall forces than attempting to measure them directly. However, if unsteady processes are under study, the local acceleration field is required to be able to perform such a prediction (Jakobsen et al. 1997; Liu and Katz 2003). Moreover, apart from the direct study of this force balance, the knowledge of the acceleration field can provide important information on the dynamics of the vortical structures present in turbulent flows (Lehmann et al. 2002; Christensen and Adrian 2002). More fundamental studies of the acceleration statistics (Hill and Thoroddsen 1997; Pinsky et al. 2000) have been performed in locally isotropic and homogeneous turbulence.They took advantage of the fact that local isotropy and homogeneity considerably simplify the theoretical formalism and reduce the number of required quantities to evaluate both the Lagrangian and the local acceleration.

Nevertheless, from an experimental point of view, difficulties of measurement can arise. On the one hand, the measurement technique used to evaluate the acceleration within the flow must have a temporal resolution in agreement with the smallest time scales of the flow. For that purpose, Hill and Thoroddsen (1997) used hot-wire anemometry to measure the fluid-particle acceleration by assuming local isotropy of the turbulent flow. The use of Taylor’s hypothesis enabled them to convert time derivatives into spatial derivatives, i.e. ∂/ ∂t+U ∂/ ∂x=0 where U is the mean velocity in the x-direction and t is time. More recently, Lehmann et al. (2002) developed estimators of the acceleration based on the laser Doppler technique and demonstrated the viability of their technique by performing acceleration measurements in a laminar stagnation point flow and in a turbulent jet. On the other hand, in order to evaluate forces or instantaneous pressure distribution by integrating the fluid-particle acceleration, simultaneous multi-point measurements are required. Consequently, previous techniques can no longer be used. To overcome this drawback, acceleration measurement techniques using particle image velocimetry (PIV) have been developed. As a matter of fact, PIV has proved to be a reliable and efficient technique to investigate fluid flows by providing two dimensional velocity fields discretized on a mesh which depends on the experimental setup and algorithm used (Westerweel 1997).

Acceleration fields can indeed be evaluated from subsequent measured velocity fields by using finite difference schemes. For that purpose, it is obvious that the acquisition rate of the PIV velocity fields must be compatible with the characteristic time scales of the flow. Therefore, the use of a single classical PIV system is limited to very slow flows, the highest frequency of which being smaller than ≃ 15 Hz. To overcome this limitation, Jakobsen et al. (1997) designed a four CCD chip camera system to acquire up to four frames in a fast time sequence. Nonetheless, performances of this setup were limited by the scanning rate of the sole laser used and the use of beam splitters to provide the light collected by the lens to the four CCD chips. Later, Dong et al. (2001) developed an algorithm that uses combination of cross-correlations and auto-correlations between pairs of doubly exposed images of the particle-seeded flow. Once again, the attainable frequencies were limited by the illuminating rate of the laser sheet generator that is used. Moreover, the computation of the acceleration from velocity fields obtained by auto-correlating a doubly exposed frame can reduce the accuracy of the displacement determination. Lourenco et al. (1998) proposed to use two double-pulsed laser systems coupled with two double frame CCD cameras to measure temporal velocity spectra. Thus, the velocity field computation can be performed via high accuracy cross-correlation based algorithms. Moreover, the two synchronized laser systems offer greater flexibility in the choice of the temporal parameters. The same kind of experimental setup was applied by Christensen and Adrian (2002) to a turbulent channel flow in order to study the vortical structure dynamics. More recently, Jensen and Pedersen (2004) presented two methods based either on the tracing of imaginary particles between the velocity fields or estimating the velocity gradients which are combined to produce the fluid-particle acceleration in order to optimize acceleration measurements by a two camera PIV system. Validation of these methods was only proposed in water wave flows.

From the above presented studies, it appears clearly that the best-suited technique to determine two-dimensional acceleration fields in turbulent flows is the one based on two PIV systems synchronized in time and optically separated. Support of this conclusion is given by recent studies aiming at developing multiplane stereoscopic particle image velocimetry. One of the important issues of the implementation of such a setup is the optical separation of the two PIV systems. In highly turbulent flows, the temporal shutter response of standard CCD cameras is much larger than the required time delay between the two PIV systems to compute time derivatives. However, each stereo camera pair must see the scattered light from only one of the two laser but has not to be illuminated by the other laser. Consequently, care must be taken to prevent such a problem and enable measurements both at the same location and the same instant. Kähler and Komopenhans (2000) first described the basic principles and the experimental requirements of the multiplane stereoscopic PIV technique. The proposed method relies on the use of two independent stereoscopic PIV systems to provide three-component velocity fields in two parallel light sheets. For this purpose, and to make it possible that each stereo camera pair is illuminated by only one light sheet, the two light sheets are orthogonally linearly polarized and the light scattered by the particles of seeding is separated by a polarizing beam-splitter cube mounted onto each CCD camera. Such an arrangement enables the separation of the two planes of measurement both in space and time. The same technique was used by Kähler et al. (2001) to investigate the spatio-temporal organization of the near wall turbulence of a turbulent boundary layer. Hu et al. (2001) measured all the components of both the velocity and the vorticity by means of an experimental setup based on the same principles. More recently, Mullin and Dahm (2005) proposed to perform the light sheet separation by the use of lasers of different frequencies instead of using orthogonal polarization. Consequently, as the optical separation no longer relies on the conservation of the polarization by the scattering particles used to seed the flow, this multiplane stereo PIV arrangement can be employed in reacting flows seeded with solid metal oxide particles that are not spherical.

These studies have demonstrated the viability of the multiplane technique for measuring gradients in turbulent flows. At this point, it should be noted that Time-Resolved PIV may now provide an alternative method to measure velocity fields in a section of the flow at high rate. Nevertheless, this new PIV technique still suffers from strong limitations: the limited framing rate prevents the investigation of highly turbulent flows and the lower spatial resolution makes velocity measurement errors large.

The objective of the present study is to extend the previous works based on two independent PIV systems, to a stereoscopic configuration in order to provide measurements of the three components of the instantaneous Eulerian acceleration in a cross section of a highly turbulent flow. Measurements are performed in a highly constraining configuration in which the largest velocity component is normal to the light sheets. Furthermore, an analysis of the differencing process is presented to provide guidelines for the determination of the best temporal parameter used to synchronize the two PIV systems. After a brief description of the flow on which the method will be tested, the experimental setup and the methodology to determine the best-suited temporal parameter to compute the Eulerian acceleration by finite difference are described in detail. The proposed method is finally applied to provide acceleration fields in a cross-section normal to the mean flow in a turbulent plane mixing layer, focusing on the dynamics of the large scale structures of the flow.

2 Test-flow: turbulent plane mixing layer

To demonstrate the viability of our approach to compute the temporal acceleration in turbulent flows from dual-time stereoscopic measurements, experiments were performed in a turbulent plane mixing layer developing downstream of the trailing edge of a splitter plate. This flow is briefly presented here, more details can be found in the work of Perret (2004). The experiments were carried out in a 0.3×0.3 m2 closed-loop wind tunnel. The walls of the test section were made of glass to provide optimal optical access. The coordinate system, described in Fig. 1, is x in the streamwise direction, y in the vertical direction and z in the spanwise direction. The origin is located at the splitter plate trailing edge. The high and low speed velocities of this two-dimensional subsonic turbulent mixing layer were respectively U a =35.2 m/s and U b =23.8 m/s, corresponding to a velocity ratio r=0.67. The boundary layers developing on each side of the splitter plate are turbulent. Their main characteristics, measured at 55 mm upstream of the trailing edge are summarized in Table 1.

Fig. 1
figure 1

Experimental setup

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Table 1 Characteristics of the turbulent boundary layers developing on each side of the splitter plate
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Particle image velocimetry measurements were performed in a cross-section normal to the mean flow located at 300 mm downstream of the trailing edge of the dividing plate, where the self-similarity region of the mean flow begins. The Reynolds number based on the local vorticity thickness δω0=18.7 mm and the mean velocity U m =(U a +U b )/2 at this longitudinal location was Re=δω0 U m /ν ≃ 36,000. Figure 2 presents the distribution of the turbulent fluctuation level 〈u ′ i 2〉1/2 across the mixing layer, where 〈.〉 denotes the ensemble average operation. It can be seen that the flow exhibits high level of turbulent fluctuations as well as strong inhomogeneity and anisotropy (Fig. 2). Velocity spectra (Fig. 3) determined from prior hot-wire measurements performed in the PIV measurements section provide information on the dynamics of the flow and enable the identification of the contribution of each temporal scale to the turbulent energy. The most energetic frequencies correspond to frequencies ranging approximately from 100 Hz to 8,000 Hz. Moreover, spectra exhibit a peak corresponding to the Strouhal number St=f δω0 /U m =0.33, which is characteristic of the presence of large scale structures in mixing layer flows. Thus, if the large scale structure dynamics is under interest, the ability of the measurement technique to correctly evaluate this frequency range will have to be checked.

Fig. 2
figure 2

Turbulent intensity of the three components of the fluctuating velocity at the measurement location

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Fig. 3
figure 3

Velocity spectra E u i u i (f) in the PIV measurements section at y/δω0=0.5 from hot-wire measurements

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3 Dual-time stereoscopic PIV setup

To perform dual-time stereoscopic PIV measurements, two SPIV systems labelled A and B, which provide two velocity fields u A(x,t A ) and u B(x,t B ) at time t A and t B , respectively, are implemented. This setup, presented in Fig. 1, is detailed in the following paragraphs.

3.1 Illumination system

To create the two light sheets used in the present dual-time PIV arrangement, two independent double-pulsed Nd:Yag lasers (Quantel, 30 mJ/pulse with an ouput wavelength λ=532 nm) were used. A combination of optical lenses was used to expand and focus laser beams into thin light sheets, the thickness of which was estimated to be 3 mm by approaching on each side a plate via a micrometer translation mechanism. This value has been chosen firstly to avoid the loss of pairs of particles due to the high out of plane velocity and, secondly, to account for the strong anisotropy of the flow in terms of fluctuation level of each component (Fig. 2). Indeed, the ratio between the root mean square of the fluctuating longitudinal velocity component and the one of each in-plane component is nearly 1.5. Thus, given the retained size of the interrogation window (1.65 mm, see Sect. 3.6), the interrogation volume satisfies an anisotropy level of the same order of magnitude. The time delay Dt between each pulse of each laser was adjustable to enable PIV processing and was controlled by the LaVision 6.2 acquisition software. Each laser sheet was linearly polarized and the polarization of the pair of pulses of one laser system was orthogonal to the polarization of the light of the other laser pulse pair. Finally, to enable precise adjustement of the position of each light sheet relative to the measurement section location, each laser system was installed on a PC-controlled longitudinal translation mechanism.

3.2 Light sheet separation

As measurements were performed in one single plane with two independent SPIV systems, the pair of cameras of one system had to be prevented from being illuminated from the light emitted by the other SPIV system. In the present study, as the two lasers were of the same type (i.e. of same wavelength), the technique proposed by Mullin and Dahm (2005) (referred to as frequency or color separation) to separate light sheets was impossible to implement. Consequently, the more widely used separating method based on the use of linearly orthogonal polarization (Kähler and Kompenhans 2000; Kähler et al. 2001; Hu et al. 2001) was employed. For that purpose, polarizing filters were mounted on each camera. The direction of polarization of the filters installed on a camera pair of one SPIV system was chosen orthogonal to the polarization direction of the second SPIV system laser.

3.3 Tracer particles

The flow was seeded with olive-oil droplets generated via a Laskin nozzle fed with pressurized air. The generated particle diameter presented a distribution with a mean diameter close of 1 μm. These tracer particles allow one to maintain the correct polarization of the scattered light. The spray generator was located downstream of the test section to ensure optimal homogeneity of the seeding in the measurement plane.

3.4 SPIV image acquisition

In order to acquire images of the seeded flow in the illuminated section, four CCD cameras were implemented in a stereoscopic configuration. Cameras used were four 1,350×1,049 CCD Intense Flowmaster cameras providing images coded with 4,096 gray levels. Acquisitions were performed in double-frame mode to allow cross-correlation PIV processing in order to obtain accurate velocity fields. As mentionned earlier, a polarizing filter was installed on each camera.

The retained stereoscopic configuration was the angular displacement configuration (Prasad 2000) with an angle about 45° between the camera axis and the normal to the light sheet (Fig. 4). Each pair of cameras within a steroscopic system was placed on one side of the measurement section, facing the other camera pair. To ensure that images were well-focused everywhere in the image plane, the Scheimpflug condition was statisfied by rotating the image plane with respect to the lens plane. To respect this condition, cameras were installed on special mounts designed to enable rotation between the lens and the CCD chip. Moreover, each camera was independently mounted on a three-translation, two-rotation mechanism to provide for easy adjustement.

Fig. 4
figure 4

Angular stereoscopic arrangement

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3.5 Temporal synchronisation

The temporal synchronization between the two SPIV systems is the key of dual-time stereoscopic PIV measurements. Consequently, an accurate timing of the whole arrangement is required. This was performed via the LaVision’s software Davis 6.2 that enables to set independently the three time delays involved in the present measurements. Indeed, for each PIV system, the time delay Dt between pulses within a pair must be determined according to the PIV requirements. Moreover, the time delay Δt between the two PIV systems must also be accurately set to enable derivative computations. The timing for recording a double pair of images is summarized in Fig. 5. The major advantage of such an arrangement is that the value of Δt can be chosen independently from the PIV parameter Dt. Hence, acceleration measurements as well as more general spatio-temporal investigations via time-correlation computations can be performed.

Fig. 5
figure 5

Timing of the cameras and laser pulses

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3.6 System calibration and image processing

The 3D-calibration method proposed by Soloff et al. (1997) was used to perform the reconstruction of the three-component vector fields from images acquired by the pair of cameras of each stereoscopic PIV system. This method enables us to take into account distortions occuring in the images due to non-uniform magnification when the angular stereoscopic configuration is used. Moreover, it allows us to avoid any measurement of the experimental setup geometry. The only parameters required to perform such a calibration are the distance between the first and second calibration planes and the distance between the crosses on the calibration target. Images of a calibration target installed on a three-dimensional micrometric translation mechanism were acquired by each camera to estimate mapping functions using a least squares polynomial approach. These polynomial functions are cubic in the y- and z-directions. In the x-direction, normal to the measurement plane, this dependence is linear. Due to the camera configuration on each side of the measurement plane, calibration of each stereoscopic PIV system was performed independently. Nevertheless, mapping functions were computed to provide vector fields onto a common spatial mesh.

Particle image velocimetry processing of the acquired particle images was performed using the LaVision’s software Davis 6.2. To improve accuracy of the vector field estimation, a cross-correlation algorithm using fast fourier transforms (FFT) was employed in conjunction with an adaptative multipass and deformed interrogation window technique. The initial size of the interrogation window was 128×128 pixels2, reduced to a final size of 32×32 pixels2 with a 50% overlapping of data. This procedure leads to a final spatial resolution of 1.65 mm. The final physical size of the measured vector fields was 75×115 mm2.

The flow parameters imposed the value of the time delay Dt between two pulses within a pair, for each PIV system. To ensure that the particle displacement does not exceed 30% of the size of the light sheet thickness (Keane and Adrian 1990), and given the fact that the maximal flow velocity is about 36 m/s, Dt was set to 20 μs.

3.7 Velocity measurement errors

The proposed method to estimate the acceleration field being based on velocity measurements, the obtained accuracy depends directly on the velocity measurement error. Parameters such as the precision of the PIV processing or error induced by the reconstruction of the three-component velocity field from two-dimensional measurements, conjugated to measurement noise inherent to any measurement technique, can alter the quality of the vector fields. The global error of measurement of the present DT-SPIV arrangement can be estimated by performing image acquisitions with a zero-time delay Δt between the two SPIV systems (Christensen and Adrian 2002). Indeed, the two light sheets being spatially superposed, in the absence of noise, velocity fields obtained from the two systems should be identical. Thus, any difference can provide information about the measurement accuracy.

3.7.1 Coincidence of the light sheets

The light sheet alignment check was performed by acquiring particle images without any polarizing filters in such a way that cameras could be illuminated by both lasers. Samples of 200 velocity fields were measured by both SPIV systems by using only one of the two lasers, with a zero-time delay Δt. With such a configuration, bad alignment between the two light sheets must lead to a systematic bias of the measured displacement by one of the pairs of cameras, leading to different statistics.

These measurements allowed computation of the probability density function of the difference between velocity components measured by the two different SPIV systems (u A i (t)−u B i (t)). Computation was performed by averaging data both in space and time to obtain statistics representative of the global error. Shape of the distribution of this velocity difference and the effect of the polarizing filters will be discussed in the next section. The constant 0.2 m/s bias in all three velocity components (Fig. 6) is attributed to the fact that the calibration of each SPIV systems was performed indepently. Given the fact that the PIV time delay Dt was set to 20 μs, this averaged bias corresponds to a displacement of 4 μm, which is of the same order of magnitude as the precision of the displacement system of the calibration target. Nevertheless, whatever the laser configuration used, no significant differences appear (Fig. 6) between velocity statistics of both pairs of cameras. Consequently, it can be concluded that the light sheets were well-superposed.

Fig. 6
figure 6

Probability density functions of difference of instantaneous velocities obtained by the two SPIV systems (u A i (t)−u B i (t)) without polarizing filters, illuminated either by the laser A (opencircle with dot at the center), or by the laser B (multiplication symbol) (Δt = 0 μs) (solid line gaussian fit). a longitudinal component, b vertical component, c transversal component

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3.7.2 Direct assessment of the measurement error.

Zero-time delay measurements performed in the configuration retained for acceleration estimation (namely by using polarizing filters and the two lasers) provide direct assessment of the global velocity error. Ideally for zero-time delay (Δt=0), the mean square error between the velocity fields obtained by the two SPIV systems 〈(u A i (x,t)−u B i (x,t))2〉1/2 should be zero. However, this quantity is not zero, because of the presence of noise that can be due to sub-pixel estimation error, electrical noise, pixel-locking error or non-uniform particle distribution within the measurement field (Christensen and Adrian 2002). Consequently, the measured velocity field can be written as:

$$ \tilde{u}_i({\bf x},t) = u_i({\bf x},t) + n_i({\bf x},t) $$
(2)

where u i is the best estimate of the velocity by the PIV technique and n i the sum of all the contributions to the noise listed above. By taking into account the characteristics of the different noise components, Christensen and Adrian (2002) showed that the root-mean-square of the fluctuating random and bias noise n i can be evaluated from zero-time delay measurements and is given by

$$ \sigma_{ni} \simeq \frac{1}{\sqrt{2}} \left\langle \left(u^A_i({\bf x},t)-u^B_i({\bf x},t)\right)^2 \right\rangle^{1/2} $$
(3)

Spatial distributions of the mean-square-error of the velocity obtained in the present study (Fig. 7 left) exhibit largest levels where the turbulent intensity is high, i.e. near the center of the mixing layer (Fig. 2). It can be noticed that error distributions of the longitudinal and transversal components u 1 and u 3 are very close whereas the vertical component error exhibits a different shape. This can be attributed to the fact that the out-of-plane and the horizontal in-plane components, u 1 and u 3, respectively, are directly linked to each other through the projection algorithm used for the reconstruction of the three component velocity field from the two displacement fields acquired by each camera. The same behaviour is retrieved when examining the probability density functions of the velocity difference (u A i (x,t)−u B i (x,t)) (Fig. 7 right). The comparison of these functions to a normal distribution confirms the gaussian feature of the noise that affects SPIV measurements. The statistical characteristics of the zero-time delay velocity difference obtained from a gaussian fit for the three components are summarized in Table 2. It shows the presence of a systematic displacement of the order of 0.04 pixel, which may be attributed to a slight misalignment between the two laser sheets and the calibration target, as pointed out in the previous section. The non-zero rms is due to the measurement noise and remains lower than 0.5 m/s (0.09 pixel). In spite of the large out-of plane velocity component in the present configuration that can make velocity measurements more difficult, these error levels are in the order of magnitude of those reported by Raffel et al. (1998) and Christensen and Adrian (2002).

Fig. 7
figure 7

Left Spatial distribution of the mean square error of between the displacement obtained by the two pairs of cameras A and B when Δt=0 μs. Right probability density function of (u A i (x,t)−u B i (x,t)); a, b longitudinal component u 1; c, d vertical component u 2 and e, f transversal component u 3. open circle with dot at the center experiment, solid line gaussian fit

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Table 2 Zero-time delay velocity difference characteristics obtained from a gaussian fit of the probability density functions of Fig. 7 right
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Moreover, it should be noted here that velocity fields acquired when using polarizing filters and the two lasers flashing at the same time (Δt=0) (Fig. 7, right) show statistics in agreement with previous results (Fig. 6), confirming that adding filters does not affect the quality of the measurements.

4 Acceleration measurement

The above presented dual-time stereoscopic PIV system was used to measure temporal derivatives of the velocity field by introducing a time shift Δt between the two synchronized PIV systems. In the following, the issue of the influence of both the measurement noise and the derivative scheme is addressed. The proposed analysis, conducted on prior measurements that are well-resolved in time, is then used to guide the choice of the best temporal parameter Δt to estimate the temporal derivative of the velocity. Eventually, examples of instantaneous acceleration field and acceleration statistics obtained in the measurement section are provided.

4.1 Influence of the derivative scheme

Computation of temporal derivatives from PIV measurements can be affected not only by the noise of measurement inherent to the PIV technique but also by the influence of the chosen derivative scheme. In fact, numerically computing derivatives introduces a truncation error due to the limited number of points taken into account but also a bias induced by the response of the derivative scheme to the noise. These characteristics were pointed out in works dealing with spatial gradient estimation from PIV measurements (Acosta et al. 2002; Raffel et al. 1998). In this context, Foucaut and Stanislas (2002) recently investigated the response to noise of different derivative schemes and proposed a methodology to choose the best-suited scheme, depending on the characteristics of the acquired vector fields. As far as temporal derivatives are concerned, Boillot and Prasad (1996) identified the main error sources that arise in PIV when one tries to evaluate the velocity field by finite difference between two particle images separated by a time Dt. To optimize this time delay, they proposed a method that takes into account the influence on the velocity measurement error of the existing random noise in the recording but also the effect of the flow acceleration within the measurement domain. Jensen and Pedersen (2004) proposed a study of the influence of the measurement noise on the acceleration evaluation, based on the analysis of the error introduced by the use of a second order finite difference scheme. An estimation of the global error in the acceleration measurement was proposed by Christensen and Adrian (2002). They also proposed a procedure based on zero-time delay measurements to correct acceleration statistics. Nevertheless, these authors did not take into account the bias introduced by the derivative scheme.

An analysis of the effect of using finite difference to compute derivatives in the temporal domain is proposed. Only the case of the second-order centered difference scheme in presence of a measurement noise considered as random is detailed, as it is the only one that can be employed from DT-SPIV measurements.

In order to improve the accuracy of the estimation of the acceleration field, a second order finite difference scheme is used to compute the acceleration at time t+Δt/2 from the velocity fields acquired at times t and t+Δt (Wereley and Meinhart 2001):

$$ \left. \frac{\partial u}{\partial t} \right|_{t+{\Delta t/2}} \simeq u^\prime(t+\Delta t/2) = \frac{u(t+\Delta t)-u(t)}{\Delta t} + \frac{(\Delta t)^2}{24} \left. \frac{\partial u^3}{\partial t^3} \right|_{t+{\Delta t/2}} + \cdots $$
(4)

Δt being the time delay between the two vector fields. In this case, the truncation error due to rejected higher order terms is of order Δt 2. If measurements are now considered to be affected by the presence of a white noise n i , the approximated derivative \(\tilde{u}^\prime\) of \({\partial u}/{\partial t}\) is now given by:

$$ \tilde{u}^\prime(t+\Delta t/2) = \frac{u(t+\Delta t)-u(t)}{\Delta t} + \varepsilon \frac{\sigma_{ni}}{\Delta t/2} $$
(5)

σ ni being the level of the measurement noise and ε the noise amplification coefficient of the derivative scheme. The parameter ε depends on the order and the nature of the considered scheme. In the present case, ε=0.71 (Foucaut and Stanislas 2002).

The influence of this discrete differencing operation in presence of noise can be studied by analysing the error between the exact derivative ∂u/∂t and the measured one \(\tilde{u}^\prime.\) For that purpose, as proposed by Lele (1992), the Fourier transform of the velocity \(u=\hat{u}\exp(i2\pi f t)\) is introduced in equation 5, where \(\hat{u}\) is the turbulence intensity at a given frequency f:

$$ \tilde{u}^\prime = \left(\frac{i\hat{u}}{\Delta t/2} \sin(2\pi f\Delta t/2) + \frac{\varepsilon\sigma_{ni}}{\Delta t/2} \right) \exp\left({i2\pi f t}\right) $$
(6)

Moreover, in the Fourier domain, the exact time derivative is directly obtained by \(\partial u/\partial t = i\hat{u}2\pi f \exp(i2\pi f t).\) Thus, for each Fourier mode, the modulus of the error \((\partial u/\partial t-\tilde{u}^\prime)/(\partial u/\partial t)\) is given by

$$ \frac{|{\rm Err}(f)|^2}{|A(f)|^2} = \left(1-\frac{\sin(2\pi f\Delta t/2)}{2\pi f \Delta t/2}\right)^2 + \left(\frac{\varepsilon\sigma_{ni}}{2\pi f \hat{u} \Delta t/2}\right)^2 $$
(7)

where |A(f)| is the modulus of the exact acceleration.

An example of the resulting error distribution with a ratio between the noise level and the turbulent intensity of 0.13, corresponding to the value obtained in the center of the mixing layer, is given in Fig. 8. It should be noted that the finite difference scheme affects the high frequencies, in the same manner as a low-pass filter with a cut off frequency f c=1/Δt, whereas the presence of noise induces a strong bias in the low frequency range. Similar results have been obtained by Acosta et al. (2002) when studying adaptative linear filters for the computation of spatial derivatives. As a consequence, these characteristics must be taken into account when designing an experiment to measure temporal derivatives and one must be aware of the spectral content of the obtained acceleration compared to the true quantity.

Fig. 8
figure 8

Error distribution of the second-order centered difference scheme in the presence of a noise obtained from Eq. 7

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Nevertheless, on the basis of the previous analysis and providing the knowledge of the PIV measurement noise level, characteristics of the acceleration by DT-SPIV can be predicted from prior measurements of the velocity well-resolved in time (by hot-wire or LDV for instance). From such measurements, the well-estimated spectra of the acceleration can be directly computed by multiplying the velocity spectra by (2π f)2. Finally, the expected error on the acceleration spectra is obtained by applying the above obtained error estimation. Therefore, it enables the exploration of a wide range of different time delays Δt in order to select the best in terms of restituted energy and frequency bandwith.

4.2 Time delay determination

The methodology described in the previous paragraph is now applied to estimate, from the hot-wire measurements presented in Sect. 2, the acceleration error spectra that can be expected when computing time derivatives from dual-time SPIV measurements.

Error distributions of the acceleration of the longitudinal component computed for various time delays are presented in Fig. 9. The parameter Δt being common for the measurement of three components the retained noise level corresponds to the worst noise level obtained in Sect. 3.7 from zero-time delay measurements and is equal to 0.07 pixel. As found by Boillot and Prasad (1996), increasing Δt leads to a better estimation of the low frequencies, which correspond to the larger scales of the flow. Conversely, a decrease of the time delay improves the prediction of the high frequency range but dramatically affects low frequencies. Consequently, the choice of Δt results from a compromise between the estimated bandwith, and the frequency range to be well-estimated.

Fig. 9
figure 9

Predicted acceleration error distribution for Δt=40, 60, 80, 100 and 160 μs with a constant noise level equivalent to an error of 0.09 pixel, at y/δω0≃ 0

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In order to determine the optimum time delay, the error distribution |Err(f)| has been integrated over the frequencies ranging from 0 to the cut-off frequency f c imposed by the second order finite difference scheme. Thus, the influence of the derivation process in presence of noise over the whole frequency range is taken into account. The computation of this global error (Fig. 10) shows the existence of an optimal time delay. Below this value, the error is mainly due to the influence of the measurement noise whereas, for larger time delay, the error of truncation becomes predominant. Therefore, from these considerations, the chosen time delay is Δt=80 μs in the present study, leading to a good prediction of the dynamics of the structures corresponding to the Strouhal frequency and to the beginning of the inertial range.

Fig. 10
figure 10

Non-dimensional integrated error between exact and estimated acceleration, accounting for both the influence of finite differencing and noise, for different locations across the mixing layer. a longitudinal component, b vertical component, c transversal component

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4.3 Acceleration measurement

Figure 11 presents the statistics of the time derivative evaluated by the proposed DT-SPIV technique with the time delay Δt chosen in Sect. 4.2. These statistics were corrected using the technique proposed by Christensen and Adrian (2002) by taking into account the measurement noise obtained by zero-time delay measurement:

$$ \left\langle \frac{\partial u_i}{\partial t} \right\rangle^2 = \frac{1}{\Delta t} \left\langle \left( u_i^B({\bf x},t+\Delta t)-u_i^A({\bf x},t) \right)^2 \right\rangle - \frac{2}{\Delta t^{2}} \sigma_{ni}^2 $$
(8)

The root mean-square of the three components of the acceleration field are compared to those predicted from hot-wire measurements as proposed in Section 4.2 using Eq. 4. Despite the uncertainty that may affect the hot-wire measurements, the predicted acceleration statistics and those directly measured by DT-SPIV are in agreement (Fig. 11). Differences that can be observed near the center of the mixing layer can be attributed to the difficulties of performing acceleration measurements in region of high shear and high level of turbulence (Fig. 2), by both techniques. Moreover, the correction proposed in Eq. 8 only takes into account the error due to the measurement noise but not the other errors inherent to the PIV technique such as peak-locking error, error due to the loss of particles or gradient error (Keane and Adrian 1990). In particular, the limited spatial resolution of the PIV measurements filters out the small scales, a bias that is less important in the hot-wire measurements.

Fig. 11
figure 11

Acceleration root mean square 〈(∂u i /∂t)2 〉1/2 (open circle with dot at the center) predicted from hot-wire measurements by taking into account the influence of the derivative scheme (Eq. 4) and (plus symbol) measured by DT-SPIV with Δt=80 μs corrected by Eq. 8. a Longitudinal component, b vertical component, c transversal component

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Examples of obtained velocity fields separated by Δt=80 μs and the resulting acceleration field are presented in Fig. 12. Instantaneous velocity fields measured in the cross section of the plane mixing layer exhibit vortical structures corresponding to longitudinal vortices typical of the secondary organization in such a flow. These structures are correctly measured by the DT-SPIV setup (Fig. 12a, b) and their temporal evolution during the time delay Δt can be seen on the corresponding acceleration field (Fig. 12c). These results demonstrate the viability of our approach to determine the best time delay to compute time derivatives by using finite difference schemes in presence of a given noise level, but also confirm the ability of the dual-time SPIV technique to provide estimation of temporal derivatives of velocity in highly turbulent flows.

Fig. 12
figure 12

Examples of instantaneous in-plane (vectors) and out of plane (gray scale contours) components of velocity field obtained by DT-SPIV: a instantaneous velocity field, b instantaneous velocity field separated from a by Δt=80 μs, c acceleration field \(\tilde{u_i}^\prime\)

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5 Conclusion

In this study, three-component acceleration measurements by dual-time stereoscopic PIV have been reported. The experimental setup, based on the use of two stereoscopic PIV systems synchronized in time, has been described. Such a PIV arrangement has been used to estimate by finite difference bidimensional three-component acceleration fields in turbulent plane mixing layer configuration, from pairs of velocity fields separated in time by a short delay. Examination of the velocity fields and the corresponding acceleration has demonstrated the ability of the proposed method to capture the dynamics of the large scale structures, given the presence of measurement noise and the use of a second-order finite difference scheme.

In order to provide guidelines for the determination of the best-suited time delay used in the differencing scheme, a method has been proposed, based on the analysis of previous well-resolved in time measurements. The computation of the spectral error distribution of the differencing process allows us to take into account both the truncation error involved in the differencing operation and the noise influence. Once this error distribution calculated, temporal spectra of the time derivative can be estimated for various time delays. Thus, the influence of this parameter on the restituted energy and frequency bandwith of the resulting derivative can be studied. Agreement between the predicted statistics and those obtained by direct DT-SPIV measurements has been shown. This methodology has been presented in the case of a second-order finite difference schemes but the same approach can be conducted on higher order scheme and then could be applied to temporal derivative computation from Time-Resolved PIV measurements.

The present method to calculate temporal derivatives has already been successfully employed (Perret et al. 2005) to derive POD based low order dynamical systems, the parameters of which being computed from samples of velocity and corresponding acceleration fields (Perret et al. 2006).