Background-oriented schlieren (BOS) techniques

The BOS technique is similar to the human observation of heat haze, mirage or fata morgana, where local density gradients between the human observer and a distant object distort or even mirror the perceived image (Fig. 1a). A second effect, the shadow effect (Fig. 1b), is visible to the unaided eye, but became more relevant with the advent of optical instrumentation. This effect is generated by a density variation between the light source and a surface of homogeneous brightness and color. The light intensity variations ∆I at each location of the surface are proportional to the second derivative of the air density between the light source and the surface. An image of these inhomogeneous light intensities is called a shadowgraph.

The observation of density gradients in fluid dynamics became important when aerodynamics became transonic. A major driver for improvement and application was the photography of bullets and their flow fields in the work of Ernst Mach at the end of the nineteenth century. The technique most commonly used for fluid density visualization is Toepler’s schlieren photography (Fig. 1c). Its optical setup typically consists of spherical mirrors or lenses and a half aperture, e.g., in form of a knife edge. It captures light intensity variations ∆I proportional to the first derivative of the fluid density.

The closest relative of the BOS methods is laser speckle interferometry (Fig. 1d). The illumination of a screen through the density gradient under investigation is made with laser light that generates a speckled interference pattern. The displacement of this pattern ∆s, which has been evaluated optically and/or digitally, is proportional to the first derivative of density.

The background-oriented schlieren method is also based on the analysis of image displacements. However, the structures are a feature of the background that can be illuminated with incoherent light and imaged through a fluid containing spatial density gradients (Fig. 2). Since digital evaluation and white-light illumination (at least for surface deformation measurements) are also known in speckle interferometry, BOS could also be named “white-light speckle density photography.” However, the name background-oriented schlieren technique is the most common name, as “background oriented” describes the fact that the camera used for the recording focuses onto objects behind the flow under investigation. Additionally, the term schlieren, the old German word for local optical inhomogeneities in (mostly) transparent media, most intuitively tells a fluid experimentalist that the technique quantifies the first spatial derivative of density, integrated along the path of light. In this article, the term “background-oriented schlieren” is always completed by the words “method” or “technique” as the “schlieren” from their definition cannot be background oriented, but only the measurement technique for their visualization.

The first descriptions of the principle and application of the background-oriented schlieren method can be found in publications from 2000. The article from Dalziel et al. (2000) appeared in “Experiments in Fluids” in April and describes several schlieren or moiré techniques that the authors named “synthetic schlieren.” They had in common that parts of the traditional optical setups have been substituted by digital image analysis. Some of these techniques and especially the variant “dot tracking refractometry” basically equal BOS. The application described within this context was the internal wave field in an oscillating cylinder. A second “Experiment in Fluids” article from Raffel et al. (2000a, b) appeared in May. It focused on the BOS principle that utilizes a random dot pattern in the background and described its application to a rotor wake visualization of a helicopter in hovering flight. The patent application of Meier (2002) was published in June. It was submitted on the same day as the publication of Raffel et al. but one year after the manuscript of Dalziel et al. (2000). The application illustrated in the patent application is the diamond-like pattern of a supersonic free jet, which had been recorded by H. Richard. It is also one of the applications described in a conference paper by Richard et al. (2000), which was presented in Lisbon in July 2000. Also described herein is the experimental and computational effort required for subsequent quantitative determination of density. Raffel et al. (2000b) presented in the same year in August in Edinburgh and described applications of the conventional, reference-free and natural background versions of BOS to a large utility helicopter in flight.

The background-oriented schlieren technique is based on the relation between the refractive index of a fluid and its density, given by the Lorentz–Lorenz equation, which can be simplified to the Gladstone–Dale equation for gaseous media. The BOS technique can best be compared with laser speckle density photography as described by Debrus et al. (1972) and Köpf (1972) and the improved versions by Wernekinck and Merzkirch (1987) and Viktin and Merzkirch (1998). Like most interferometric techniques, laser speckle density photography preferably utilizes an expanded parallel laser beam, which traverses a compressible flow field or (in more general terms) passes through an object of varying refractive index (i.e., a phase object). However, in contrast to other interferometry, laser speckle patterns are generated instead of interference fringes. Compared with the quantitative optical techniques mentioned above, the BOS method simplifies the recording. The laser speckle pattern, usually generated by the expanded laser beam and a ground glass, is replaced by a (mostly painted or printed) random dot pattern on a surface in the background of the test volume. This pattern preferably has to have the highest spatial frequency that can be imaged with sufficient contrast.

Usually, the recording is performed as follows: First, a reference image is generated by recording the background pattern observed through air at rest in advance of (or subsequent to) the experiment. In the second step, an additional exposure through the flow under investigation (i.e., during a wind tunnel run) leads to a locally displaced image of the background pattern. The resulting images of both exposures can then be evaluated by image correlation methods. Existing evaluation algorithms, which have been developed and optimized, for example, for particle image velocimetry (or other forms of speckle photography), can then be used to determine the displacement of patterns at multiple locations throughout the image. The deflection of a single beam contains information about the spatial gradient of the refractive index integrated along the line of sight (Fig. 3). Details on the theory of ray tracing through gradient-index media are found in Sharma et al. (1982) and Doric (1990).

Assuming paraxial recording and small deflection angles (ɛ _y  ≈ tan ɛ _y ), a formula for the image displacement Δy can be derived for the BOS technique (Raffel et al. 2007) with the magnification factor of the background, M = z _i /Z _B, the distance between the dot pattern and the density gradient, Z _D, and the deflection angle

The image displacement can thus be rewritten aswith Z _A being the distance from the lens to the object and the focal length of the lens, f. Since the imaging system should be focused onto the background for best contrast, we note:

Equation 3 shows that a large image displacement can be obtained for a large Z _D and a small Z _A. The maximum image displacement for increasing Z _D approaches \(\Delta y = f\varepsilon_{y}.\) On the other hand, certain constraints in the decrease in Z _A have to be fulfilled in order to image the flow field with sufficient sharpness. The optical system has to be focused on the background in order to obtain maximum contrast at high spatial frequencies for later interrogation, and Eq. 4 applies. At the same time, the sharp imaging of the density gradients would be best at \(Z_{\text{i}}^{{\prime }}\) with

This focusing problem that is inherent with the BOS technique is depicted in Fig. 4. As a consequence of the fact that the system is usually focused on the background, the sharpness of the density gradient under investigation is limited. By introducing the aperture diameter d _A and the magnification of the density gradient imaging \(M^{\prime} = Z_{\text{i}}^{{\prime }} /z_{\text{A}},\) a formula for the geometric blur d _i (Fig. 4) of a point at Z _A can be expressed as:

Additionally, the imaging of the small scales structures in the background is diffraction limited. The following formula holds for the diffraction limited minimum image diameter d _d.where λ is the wavelength of the light (~0.5 μm). The approximation given by (Eq. 8) can be used in order to compute the overall image blur \(d_{\varSigma }\) for the two different sources and to simulate the effect of the underlying convolution of the imaging artifacts:

The following problem arises when trying to optimize the sharpness of BOS recording by minimizing the overall image blur \(d_{\varSigma }\): Larger aperture diameters d _A will decrease the diffraction limited minimum image diameter d _d (Eq. 7), but increase the geometric blur d _i (Eq. 6). However, for most BOS setups, the latter effect is the stronger resulting in small aperture diameters typically being used for BOS recording. This increases the demand for intense background illumination, but also keeps other imaging problems like spherical and chromatic lens aberrations small. Since correlation techniques average over the interrogation window area, the overall image blur \(d_{\varSigma }\) does not lead to a significant loss of information, as long as \(d_{\varSigma }\) is considerably smaller than the interrogation window size.

The following section describes a typical BOS experiment. More complex experimental setups and evaluation methods can be found in the literature which is described in the later sections.

The opening angle of a BOS camera, determined by the focal length of the lens and the sensor format, is chosen in a way that the object’s image covers the full image area. The background pattern is placed at a distance to the object, which is typically smaller than the distance between the object and the camera, but of the same order of magnitude. The f number of the imaging lens (f/d _A) is in the order of 11. The typical size of an individual background structure (e.g., a random dot) yields an image on the order of 3–5 pixels. Two- to three-pixel images are better, but can frequently not be reached, because of the diffraction limited imaging at small apertures and the small pixel pitch of today’s cameras. A recording of an undisturbed background (flow-off) and a disturbed background (flow-on) are loaded into the software used for the cross-correlation evaluation (e.g., by PIV software). The peak fitting routine should be adapted to this image size (e.g., 4 pixels). The 3-point peak fitting frequently used for PIV evaluation will yield more measurement noise, when BOS image structures are larger. Conventional cameras can be used as well as scientific cameras, but image format should be uncompressed. Demo versions of suitable software packages can be downloaded from the internet (e.g., PivView and others). The BOS image displacements due to density gradients are usually smaller than that in PIV recordings. In spite of the fact that small image displacements (e.g., <0.5 px) can be determined with high accuracy, the small displacements lead to relative BOS measurement errors that are larger than usually obtained in PIV (e.g., BOS 2–3 % instead of PIV 1 %). Larger image displacements can be obtained for smaller fields of view. The subtraction of the mean displacement helps to see the flow patterns when vector plots are used for visualization. Figure 5 depicts an undisturbed background, recorded as reference (a), the same background behind a laser-induced plasma (b) and the resulting image displacements proportional to the density gradients along the line of sight (c). The displacements are represented as vectors (Fig. 5c). The underlying grayscale represents the radial displacement component at each location. Typical effects that might occur for such thermal applications are: the image blur due the strong density gradients, the reduced image contrast behind the ionization spark and the resulting data dropout in the center of the displacement field.

In preparation for a 3D localization, the positions of the flow structures under investigation have to be determined from the evaluated displacement images to get their 2D positions in each camera plane. Bauknecht et al. (2015), for example, describe a semiautomatic detection algorithm for finding discrete points on vortex lines. Starting from manually selected points close to the vortex filaments, a stepwise cross-correlation between cuts orthogonal to the main vortex axis in combination with a polynomial-based predictor is applied. A spline curve is then fitted to the discrete points on each vortex filament by means of a least squares procedure.

For the 3D reconstruction of the geometry of a solid object with photogrammetry, discrete markers or prominent points on its surface are usually imaged and identified in multiple measurement images of a multi-camera system. In order to apply the same reconstruction algorithms to invisible and continuous objects such as vortices, corresponding points on the vortex lines have to be identified in the images of all cameras (see “correspondence problem” in Meyn and Bennett 1993).

One possible solution for this is the application of the epipolar geometry method (Hartley and Zisserman 2003), which imposes geometric constraints on pairs of camera images, as shown in Fig. 6. Epipolar geometry uses the pinhole camera model and is based on the idea that a plane defined by an object space point P (e.g., a point on a 3D vortex segment) and its projections P _A and P _B onto the image planes of two cameras (A, B) can be used to establish geometric dependence between P _A and P _B. If only P _A and the corresponding optical center O _A are known, the line through these two points can be projected as an epipolar line in the image plane of camera B (red line in Fig. 6). The projected point P _B corresponding to P _A must be located on this epipolar line. If the vortex on which P is located on can be identified in both projections, as shown in Fig. 6, the point P _B can be found by intersecting the epipolar line with the corresponding projected vortex in the image plane of camera B. The constraint imposed by epipolar geometry can be written aswith the coordinates of the corresponding points in the images A and B and the 3 × 3 fundamental matrix F. This fundamental matrix can be estimated given at least seven corresponding point pairs in the two images. In the study described by Bauknecht et al. (2015), the fundamental matrices were estimated for all possible camera combinations in the multi-camera setup.

The intersection points of the epipolar lines with the corresponding 2D vortex projections were computed and used in the reconstruction process. The 3D reconstruction of the vortex positions was carried out using pairs of cameras and the stereo-photogrammetry technique. A result is shown in Fig. 7, depicting a large part of the vortex system of a full-scale BO 105 helicopter in hovering flight.

It is generally possible to use a finite difference approximation or a Poisson solver to integrate the density gradient field to determine the relative density field in the flow. If the absolute density is known at one point in the observation area, this can be used to determine the density at all points in the flow quantitatively. However, any non-two-dimensionality in the flow field will distort the results, since the BOS method is a line-of-sight integrating technique. A first step toward the determination of the density distribution by computation of the Poisson equation for BOS data of two-dimensional transonic flows has been presented by Richard et al. (2000) and Richard and Raffel (2001). However, a more complete analysis by tomographic algorithms is required in order to deal with three-dimensional density fields.

van Hinsberg and Rösgen (2014) and Van Hinsberg (2014) used the principle of ray tracing and the ring discretization method to obtain the theoretical deflections and apparent shifts of an axisymmetric supersonic underexpanded free jet. They also described a modification of the classical correction factor for BOS applications in the near-field. The density field of axisymmetric flows can also be computed from the schlieren data after applying an Abel or Fourier transform inversion algorithm. In this case, the transform algorithm rebuilds the density field from ray deflections. This has been shown, for example, by Venkatakrishnan (2005), Venkatakrishnan and Suriyanarayanan (2009) and Sourgen et al. (2004, 2012). Kindler et al. 2007 applied the tomographic reference-free BOS method for the study of rotor blade tip vortices. Ota et al. (2011) demonstrated that the Algebraic Reconstruction Technique (ART) can also be used in order to derive density distributions. According to them, the ART method is favorable for applications where a wind tunnel model is in the line of sight.

Some of the publications on the density determination for complex flows deal with investigations of an asymmetric mixture of jets (Venkatakrishnan et al. (2011) or an asymmetric cold streak in a turbine engine’s exhaust jet (Adamczuk et al. 2013; Hartmann et al. 2015). If single jets were investigated by tomographically evaluated BOS recordings, the authors aimed for the description of the unsteady effects. Todoroff et al. (2014) reconstructed instantaneous 3D flow density fields of an unsteady air jet by a new direct regularized 3D BOS method. Atcheson et al. (2008) and Berger et al. (2009) applied tomographic BOS to unsteady gas flows.

Goldhahn and Seume (2007) and Goldhahn et al. (2009) described the tomographic reconstruction by a filtered back-projection algorithm and applied the method to determine the density distribution of asymmetric underexpanded free air jets out of a double-hole orifice. The 3D density field was reconstructed in planes perpendicular to the jets axis. They justified the assumption of parallel projection in the case of small opening angles of the camera lens used. The principle of the reconstruction process can be described as follows:

The refractive index is related to the density by the Gladstone–Dale equation:where n is the refractive index and G is the Gladstone–Dale constant. A relation between the angle of deflection and the first derivative of the refractive index was derived by Fomin (1998). By applying Eq. (2), this expression can be modified as:

In order to apply computerized tomography, secondary radial axes s and t were established on the plane YZ. Figure 8 shows these secondary radial axes. Different recording projections can be obtained by rotating the axis s by an angle θ for a range of 180°. By taking enough projections through the object, the Fourier plane is successively filled with values that describe the object. A single light ray in a line \(s = {\text{const}}\), with \(s = z\;\cos \varTheta + y\;\sin \varTheta\), will be deflected by an angle \(\varepsilon (s,\varTheta )\) (see Liu et al. (1989) for a detailed description):where the integral is calculated along a line \(s = {\text{const}}\). Equation 12 basically represents Eq. 11 adapted to the radial axes. In order to compute the density reconstruction, the convolution back-projection method can be applied (Goldhahn and Seume 2007).

Hence, by using Fourier transforms, the density distribution can be obtained,where the density reconstruction is a convolution of the projections under ɛ(s, θ) with a filter function q(s).