To avoid loss of correlation due to excessive in-plane displacement, Keane and Adrian [1] state that the displacement s should be smaller than one quarter of the interrogation window size d_I, or s = τ U / M < 1 4 d I. This yields a maximum value for the pulse separation τ for a given velocity magnitude U and pixel scaling M [m/px]. Smaller τ values result in a slightly stronger correlation; however the displacement should remain greater than the minimum resolvable displacement. Incorporating this rule, the dynamic velocity range for single-pass correlation is (2) DR V ( s ) = 1 4 d I σ s ( s )

Raffel et al. [3] and Westerweel [4] review the dependence of the total displacement error (and thus σ_s) on a number of parameters for single-pass correlation (e.g., particle displacement, number density and diameter, interrogation window size, image background noise, velocity gradients).

Westerweel et al. [5] describe a multi-pass correlation approach by shifting windows over discrete pixel amounts, based on the local displacement obtained in the previous pass. Simulation results show a threefold reduction in displacement uncertainty Δs_rms. Validation results of grid-generated turbulence in a water channel show a typical displacement uncertainty of 0.04 px, compared to 0.095 px without window shifting [5]. The technique has since been improved to continuous shifting, applying image interpolation techniques [6].

Scarano and Riethmuller [7] describe an iterative window deformation method with progressive grid refinement. Monte Carlo simulations of noiseless artificial images yield uncertainty values of about 10⁻³ px [8]. Multi-grid techniques partly decouple the maximum displacement and final window size, since the 1/4 window rule [1] only applies to the first (coarse) grid. For a progressive refinement from an initial window k_gd_I to final window d_I (k_g > 1), Equation (2) can be rewritten as: (3) DR V ( m ) = k g 1 4 d I σ s ( m )

For the same final window (d_I), DR_V increases by the grid refinement ratio (typically 2 ≤ k_g ≤ 4). A further increase is due to a reduction of σ_s. Westerweel [5] and Scarano and Riethmuller [6] report an uncertainty reduction σ_s^(s)/σ_s^(m) ≅ 3 for discrete window shifting and σ_s^(s)/σ_s^(m) ≅ 10 for subpixel window shifting and deformation, respectively. However, these values are obtained for noiseless artificial images and the uncertainty increases for more realistic conditions, e.g., non-zero gradients [9].

In the remainder of the paper, ‘conventional’ PIV refers to the current state of art multi-grid cross-correlation using subpixel window shifting and deformation.

Increasing the pulse separation to enhance the dynamic range is generally not preferred. However some studies present satisfactory results when the increase is applied locally [10–12]. These techniques use single-frame imaging, and are proposed as alternatives to multi-grid methods.

Fincham and Delerce [10] suggest a multi-frame (MF) approach based on a series of single-frame recordings, where an initial correlation of two frames (separated by inter-frame time δt) is used as a displacement estimate for the deformation and correlation of frames separated by 2δt or 3δt. This aim is to increase the average pixel displacement, thus improving the dynamic velocity range.

Hain and Kähler [11] also propose an iterative MF technique to compensate for the loss in dynamic range of CMOS sensors used in high speed PIV systems, compared to CCD sensors. On a single-frame sequence {...t − 2δt, t − δt, t, t + δt, t + 2δt...}, an initial correlation is performed on frames t − δt and t + δt. The correlation is repeated between frames t − k_τδt and t + k_τδt, where the multiplier k_τ is estimated based on the quarter window rule assumption and the local displacement s, as k τ = 1 4 d I / s. In selecting the optimal k_τ, Hain and Kähler [11] indicate that a simple threshold for the correlation peak ratio Q (i.e., ratio of highest to second highest correlation peak [1]) is not sufficient for optimality. The authors assume a minimum resolvable displacement of 0.1 px.

Multi-frame PIV is most suitable for low speed flows. Hain and Kähler [11] validate their technique with direct numerical simulations of a laminar separation bubble (U_max = 0.15 m/s) and experimental velocity data around an airfoil in water (U_max = 0.1 m/s). Pereira et al. [12] propose a similar MF technique and compare it to multi-grid PIV, for test cases including artificial particle images (U_max = 1 px/s and δt = 1 s) and a laminar water flow (U_max = 0.05 m/s). In these cases with a wide velocity range, MF PIV has achieved good results compared to conventional PIV. However, since MF PIV is proposed as an alternative to multi-grid algorithms, it cannot benefit from advances in this field.

This paper proposes a new multiple pulse separation (MPS) technique to increase the dynamic velocity range of PIV. The technique is based on double-frame imaging, thus avoiding the low speed restriction and excessive pulse separations of MF PIV [10–12]. It does not exclude the use of multi-grid algorithms. A robust criterion for pulse separation optimality is established and validated.

Consider a flow field with a wide range in velocity magnitude (e.g., a jet or wake flow), where U_max and U_min represent two characteristic velocity scales in the high and low velocity regions, respectively. As the ratio U_max/U_min approaches the dynamic velocity range of the measurement technique (DR_V), the vector quality in the low velocity region deteriorates. For this reason multi-frame correlation was first proposed [10–12]. By selectively applying a higher pulse separation k_ττ only in the low velocity region, the minimum measurable velocity reduces (σ_V ∝ σ_s/(k_ττ)) and the dynamic velocity range increases: (4) DR V ( mps ) = k g 1 4 d I τ σ s ( m ) / ( k τ τ ) = k τ ︸ pulse   separation multiplier k g ︸ grid refinement 1 4 d I σ s ( m )The increase in DR_V is proportional to the applied pulse separation multiplier k_τ, which in turn is determined by the optimality criterion described in the following section.

Contrary to the multi-frame approach [10–12], multi pulse separation (MPS) PIV acquires double-frame images {...,[t, t + k_τ,1τ], [t + δt, t + δt + k_τ,2τ],...} with N different pulse separation values k_τ,iτ (i = 1…N) at a frame rate 1/δt, where the N multipliers k_τ,1, k_τ,2, … k_τ,N represent monotonically increasing values (e.g., 1, 4, 16). Figure 1(a) depicts the conventional double-frame (single exposure) PIV approach with a single fixed pulse separation τ. The subscript j is the index in the sequence of acquired image pairs, and the subsequently evaluated displacement fields s⃗ (x,y)_j the arrow notation is often omitted hereafter). Figure 1(b) depicts the MPS PIV approach: (i) a sequence of double-frame images [I(t), I(t+τ_i)]_j is acquired, while the pulse separation loops through N chosen values (τ_i = k_τ,iτ). Next (ii) the vector fields for all pulse separation values are evaluated using conventional multi-grid algorithms. Finally (iii) the pulse separation optimality criterion (described below) is applied in a post-processing step, resulting in the final displacement fields s⃗_opt(x,y)_j.

The peak ratio Q is a measure of the correlation strength of a displacement vector [1]. To assess the local precision, the displacement magnitude |s| is compared to the minimum resolvable displacement σ_s. As a precision measure, 1 − σ_s/|s| varies between unity for |s| σ_s, over zero for |s| = σ_s to -∞ as |s| → 0. The weighted peak ratio Q’ is defined as a measure of local vector quality, combining correlation strength and precision: (5) Q ′ = Q ( 1 − σ s ‖ s → ‖ )

The pulse separation optimality criterion is based on the local maximum of Q’. In each point (x, y), the local maximum of Q’ = Q’(x,y,τ_i) = Q(x,y,τ_i) (1 − σ_s/‖s⃗(x,y,τ_i)‖) the local optimum pulse separation. The approach assumes that the value of σ_s does not vary significantly within the field of view, which is true for typical laboratory conditions with background image noise and velocity gradients. Although advanced multi-grid algorithms can attain errors below 0.001 px in noiseless conditions [6,8], a value for σ_s in more realistic conditions is about 0.1 px. In the optimality criterion, values of σ_s between 0.05 px and 0.2 px yield the best results. This order of magnitude seems appropriate for multi-grid algorithms in realistic conditions, based on validation results in the review of Stanislas et al. [13].

A selector operator is defined based on the maximum Q’ value: (6) for   any   variable   a i ( τ i ) : sel Q ′ ( a i ) = a i | Q i ′ = max i ( Q i ′ )

The optimal pulse separation, displacement and velocity fields are determined as (7) τ opt ( x , y ) = sel Q ′ ( τ i ) = τ i | Q i ′ = max i ( Q i ′ ) s → opt ( x , y ) = sel Q ′ ( s → ( x , y , τ i ) ) U → opt ( x , y ) = M   s → opt ( x , y ) τ opt ( x , y )

Based on Equation (7), each vector is taken from a single measurement according to the local maximum Q’ value. An alternative definition is based on a linear combination, weighted according to the value of Q’. A relaxed maximum selector is therefore defined as (8) for   any   variable   a i ( τ i ) : sel Q ′ ( p ) ( a i ) = ∑ a i w i ∑ w i with   w i = [ Q i ′ − min i ( Q i ′ ) max i ( Q i ′ ) − min i ( Q i ′ ) ] pwhere p > 1. As p → ∞, the weights tend to w_i = 1 for Q′_i = max_i Q′_i and w_i = 0 otherwise, and the relaxed maximum selector reverts to Equation (6), or lim p → ∞ ( sel Q ′ ( p ) ( a i ) ) ≡ sel Q ′ ( a i ).

Using Equation (8) the optimal displacement, velocity and pulse separation are (9) s → o p t ( x , y ) = sel Q ′ ( p ) [ s → ( x , y , τ i ) ] U → o p t ( x , y ) = sel Q ′ ( p ) [ U → ( x , y , τ i ) ] τ o p t ( x , y ) = M ‖ s → o p t ( x , y ) ‖ ‖ U → o p t ( x , y ) ‖

The optimality criterion based on the relaxed maximum (Equation (9)) yields smoother results since data obtained at different pulse separations are combined, weighted by the local Q’ value. The exponent p determines the relative contribution of data obtained at sub-optimal pulse separations. In practice p = 5 yields good results, while the difference between Equations (7) and (9) is negligible for p > 20.

In choosing the pulse separation multipliers k_τ,i (i = 1…N), the smallest value τ (k_τ,1 = 1) should limit the correlation loss in the high velocity region, based on e.g., the 1/4 window rule [1] or similar considerations. The maximum k_τ,N can be chosen analogously for the low velocity region, e.g., as. k_τ,N ≅ U_max/U_min Regarding the total number of values, N = 2 or 3 typically yields good results while limiting the additional acquisition and processing time.

Compared to conventional PIV, the maximum increase in dynamic velocity range is DR V ( mps ) / DR V ( m ) = k τ , max (see Equation (4)), where k_τ,max < k_τ,N since the optimality criterion does not necessarily select the largest applied pulse separation. From Equation (4), the actual dynamic velocity range for MPS PIV is given by (10) DR V ( mps ) = k τ k g 1 4 d I σ s ( m ) with   k τ = max x , y [ τ opt ( x , y ) ] τwhere τ_opt(x,y) follows from Equation (9). Depending on the flow conditions and the value of the minimum resolvable displacement σ_s, the dynamic velocity range can increase by more than one order of magnitude compared to conventional multi-grid PIV, as shown in the validation results in Section 3.2.

Mann and Picard [14] introduced a technique to combine photographic images with different exposure times, to extend the dynamic intensity range beyond the restrictions of a digital sensor. A composite high dynamic range (HDR) image is generated as the weighted sum of all images. Weighting or ‘certainty’ functions are determined to favour mid-range intensity values, corresponding to the maximal sensor sensitivity and avoiding clipping near the edges of the range. Reinhard et al. [15] and Battiato et al. [16] discuss several weighting approaches to match the nonlinear response curve of an optical sensor array.

The MPS PIV technique proposed in this paper shows some analogies to HDR imaging. In both cases, a high dynamic range composite field is generated from a set of low dynamic range fields with different ‘exposure times’. Similarities persist in the optimality criterion used to construct the composite field. In HDR imaging, continuous weighting functions are used to provide a gradual transition between dark (underexposed) and bright (overexposed) regions. Thus each pixel contains information from all images in the set. MPS PIV also uses continuous weighting functions based on the local weighted peak ratio Q’, given by the relaxed maximum criterion (Equations (8) and (9)). However, a high exponent (p ≅ 5) is applied in Equation (8) to limit the contribution of data obtained at sub-optimal pulse separation values. In the extreme case where p → ∞, Equation (8) tends to Equation (6) and becomes a strict maximum selector, where each vector is selected from a single pulse separation acquisition.

The contribution of data from sub-optimal pulse separations should be limited in MPS PIV due to the strongly nonlinear nature of the correlation peak detection in PIV. Spurious vectors for excessive pulse separation values must not be allowed to propagate into the composite velocity field. As for any other technique, MPS PIV should be applied with good judgment.

Similar to conventional PIV, MPS PIV is applicable to stationary or non-stationary flows. MPS PIV correlates double-frame images separated by τ_i, whereas MF PIV correlates single-frame images separated by multiples of δt. Determined by the system repetition rate f_F, the minimal δt far exceeds the minimum pulse separation for double-pulsed systems (τ 1/f_F,max ≤ δt). This is a significant distinction between MPS and MF PIV. Excessive particle displacement limits MF PIV to low speed flows. Hain and Kähler [11] and Pereira et al. [12] report maximum velocities below 0.1 m/s in practical applications. Using double-frame imaging, MPS PIV is applicable to low and high speed flows in the same way as conventional PIV.

For temporal or spectral analyses, the common limitation for MPS and MF techniques is that a single recording duration Nδt (or N/f_F) should be smaller than the flow time scale, where N is the number of pulse separations. The same restriction applies to conventional PIV, albeit for N = 1.

For amplitude domain analysis, no restrictions apply for single-point statistics (e.g., mean, variances and Reynolds stresses, higher order moments, probability density functions). For two-point statistics (e.g., spatial correlation) only point pairs acquired at the same measurement time should be considered.

MPS PIV is not an alternative but an addition to multi-grid techniques, without restricting the use of advanced methods such as window shifting and deformation. For the validation results (Section 3), the technique is implemented as a set of macro functions in LaVision Davis 7.2.2, using its multi-grid algorithms with deformation for vector evaluation.

A single round stationary jet of air impinges perpendicularly onto a flat surface (Figure 2). The orifice diameter D = 5 mm and the orifice-to-surface distance H = 4D. The axial and radial coordinates x and r are aligned along the jet axis and perpendicular to it, respectively. The jet issues from a straight-edged orifice of length 2D, connected to a settling chamber. The velocity distribution in the orifice is axisymmetric yet not radially uniform. This is not important for the test case, and no effort was made to prevent flow separation at the upstream orifice edge. The flow rate is measured and maintained constant using a digital mass flow controller (MKS 1579A, 300 standard litres/min, repeatability ±0.2%). All experiments are performed at a fixed Reynolds number of Re = 8,000, based on D and the mean velocity in the jet orifice (U_m = 24 m/s). Fitzgerald and Garimella [17] present velocity distributions measured using LDV in a similar geometry for Re = 8,500.

Figure 2 identifies four distinct regions in the flow field: (i) the free jet with a decaying potential core in the centre and surrounding shear layer, (ii) the stagnation region, (iii) the wall jet and (iv) the entrainment region. Each of these features a significantly different characteristic velocity magnitude, making this an interesting test case for the proposed methodology.

The PIV system comprises a New Wave Solo-II Nd:YAG twin cavity laser (30 mJ, 15 Hz) and a LaVision FlowMaster 3S (PCO SensiCam) thermo-electrically cooled CCD camera (1,280 × 1,024 px², 12 bit) with 28 mm lens. The image magnification is 1:3.4 (M = 45 μm/px). A glycol-water aerosol is used for seeding, with particle diameters between 0.2 and 0.3 μm. The particle image diameter is adjusted to d_p ≅ 2 px by defocusing slightly. Customized optics generate a 0.3 mm thick light sheet. The CCD camera is mounted perpendicular to the light sheet. The velocity fields are processed with LaVision’s DaVis 7.2.2 software, using multi-grid cross-correlation with continuous window shifting and deformation, with a window size decreasing from 64 × 64 px² to 32 × 32 px² and a 75% overlap. The validation is based only on amplitude domain statistics (mean flow and turbulence intensities). As such, a low speed PIV system can be used in this stationary flow configuration.

The LDV system comprises a 500 mW Ar⁺ laser and a dual beam Dantec optics with 488 nm (blue) and 514 nm (green) wavelengths to measure axial (along x) and radial (along r) velocity components, respectively. The optical head applies Bragg cell frequency shifting to both components. The system is operated in backscattering mode to facilitate translation and near-wall measurements. The measurement volumes are about 0.12 mm in diameter and 1.6 mm long, with the long axis aligned in the out-of-plane (z) direction. The same aerosol seeding is used. The velocity data are evaluated using a Dantec BSA F50 burst spectrum analyser. Velocity weighting and statistics are performed using Matlab, applying inverse velocity magnitude weighting to reduce high velocity bias errors.

Figure 3(a) shows a time-averaged streamline plot for the jet flow obtained using conventional PIV. The term ‘conventional’ here denotes the best possible selection of the pulse separation τ = τ_min which maximizes vector quality throughout the field of view, and the same above described algorithm. The quarter window rule in this case suggests τ < 1 4 M d I / U max = 30   μ s, however a further reduction was needed due to strong gradients in the shear layer. To limit correlation loss due to gradients, Westerweel [9] derived a pulse separation threshold as d I | ∂ U / ∂ x | τ < 2 3 d p (for single pass correlation). For a shear layer gradient ∂U/∂x ≅ U_max/(D/2) = 9,600 s⁻¹, the threshold yields τ < 4.3 μs. In practice, a maximum value of τ (= τ_min) = 5 μs was found to ensure good vectors in the shear layer, resulting in a displacement of about 3 px in the jet core, and a gradient of ds/dr(d_I/d_p) ≅ 0.75 px/px in the shear layer. The strong gradient is the limiting factor here, yet the value of 0.75 px/px is comparable to that achieved by other authors in strong shear flows using multi-grid correlation [8]. Attempts to further increase τ (e.g., by decreasing the initial window) resulted in invalid vectors in the shear layer region.

Figure 3(b) shows the corresponding results for a 10 times larger pulse separation τ = 10τ_min, yet otherwise identical acquisition and processing parameters. In the high velocity jet core region, the streamlines break down due to the absence of valid vectors, whereas the low velocity region shows smoother streamlines for τ = 10τ_min than the ones for τ = τ_min in Figure 3(a).

The MPS technique proposes the weighted peak ratio Q’ = Q(1 – σ_s/|s|) as a measure of local pulse separation optimality. Distributions of Q’ are plotted in Figure 3(c,d). For both pulse separation values, the region of best vector quality corresponds to high values of Q’. These occur in the jet core and wall jet region for small pulse separation τ = τ_min (Figure 3(a,c)) and in the entrainment region for the larger pulse separation τ = 10τ_min (Figure 3(b,d)).

Figure 4(a,c,e) shows the corresponding MPS PIV results after applying the optimality criterion (σ_s = 0.2 px and p = 5 in Equation (8)) to the data obtained at two pulse separation values τ/τ_min = {1, 10}. Figure 4(b,d,f) shows MPS results for data acquired at seven values τ/τ_min = {1, 2, 4, 10, 20, 40, 100}.

Figure 4(e,f) shows the distribution of the optimal pulse separation τ_opt(x,y)/τ_min. The smallest values τ ≅ τ_min are used in the jet core region, and larger values τ ≅ 10τ_min in the entrainment region. When applying a larger number of pulse separations, Figure 4(f) shows that intermediate values 2 < τ/τ_min < 4 are used for the stagnation and wall jet regions, and high values 10 < τ/τ_min < 40 in the lowest velocity regions.