For the investigation of compressible flows the determination of the density distribution is of prime importance. For this purpose, the schlieren method, introduced by A. Toepler in 1864, is currently used [1]. The schlieren technique transforms the phase variation of the light passing through a phase object into an intensity variation. Other techniques such as the density speckle photography appeared in the seventies and allowed the direct measurement of the deflection of the light [2,3]. Later, Wernekink and Merzkirch [4] and Niessen et al. [5] used an improved version of the density speckle photography for this purpose.

The Background Oriented Schlieren (BOS) technique is based on a patent held by Meier [6] and more precisely described by Raffel et al. [7,8]. In order to measure the light deflection caused by density gradients in a compressible flow, the BOS technique uses the distortion of a background image. Tiny, randomly distributed dots on a flat plate are used as a background. The recording has to be performed as follows: first a reference image is generated by recording the background pattern observed through the air at rest before or after the experiment. Secondly, an additional exposure through the flow under investigation leads to a displaced image of the background pattern. The resulting images of both exposures can then be evaluated by correlation methods, leading to the local displacements related to the deviations of the rays. This paper describes how to improve the accuracy of the BOS technique and how to treat strong density gradients causing blurred images by using a background with colored dots and suitable correlation algorithms (CBOS technique).

Furthermore, the measured displacements of the rays are a projection of the gradients of the density field in a given observation direction. By using multiple projection data the three-dimensional density distribution of a flow field can be reconstructed, for example, with Computed Tomography (CT). An axisymmetric flow density distribution can be obtained with techniques based on the Abel or the Fourier transforms, as shown by Venkatakrishnan et al. [9,10], Sourgen et al. [11] and Kak et al. [12]. In the papers by Ota et al. [13,14] the Algebraic Reconstruction Technique (ART) is also shown to be suitable in the reconstruction of density fields, especially for the case where the flow around a model has to be reconstructed.

Former studies about spike-tipped models pointed out [15-18] the interaction between the incoming free stream flow and the high-pressure region near the model, which could have an influence on upstream at the tip of the spike by displacing the boundary layer along the spike and subsequently inducing a large-scale instability of the flow. In this paper, the unsteady flow around a spiketipped model at Mach number 3 is studied. Therefore a high-speed camera is used. Supposing the flow to be axially symmetrical, an algebraic reconstruction technique (ART) is applied to reconstruct the density field around the spike.

The experiments are carried out in the 0.2 m ´ 0.2 m supersonic blow-down wind tunnel of ISL with a freestream Mach number of 3. The Reynolds number Re_D based on the model diameter (D = 40 mm) is 2.7 ´ 10⁶; the tunnel freestream static pressure p is 190 hPa and the

freestream density ρ is 0.651 kg∙m⁻³. The test models used for this investigation have a cylindrical centerbody mounted on a sting assembly along the wind-tunnel centerline. The angle of attack is set to zero. The spike at the front of the model is l = 35 mm long and the nose is a cone with a half-angle of 14˚ (Figure 3).

The optical setup is built perpendicularly to the symmetry line of the model. The CBOS images are recorded with a Phantom v1610 high-speed camera with 27,000 fps (frames per second). The exposure time is 36 μs. Due to this high frequency the size of the images is 640 ´ 800 pixels. The background is illuminated by a continuous white light source. The reconstruction of the density field is based on the pictures from the high speed camera. In order to increase the depth of field and the accuracy of the CBOS measurement, a telecentric optical system (Figure 13) is introduced, as described by Ota et al. [24]. Furthermore, complementary images with higher resolu-

tion are recorded with a high-resolution camera (Canon EOS 1 Ds Mark II) with a normal lens (f = 50 mm). The CMOS sensor of the camera has a resolution of 4992 × 3328 pixels. The background is illuminated by a flashlamp for a duration of 2.5 μs [25]. In order to increase the depth of field, all pictures are taken with the smallest aperture, while the cameras are focused on the artificial background.

The Algebraic Reconstruction Technique (ART) is an iterative reconstruction method. The unknown density field is discretized and for each voxel (“volumetric pixel”) the density gradients and finally the density has to be determined. In order to solve this array of unknowns the Equations (2) have to be set for the measured displacement vectors for each projection as a function of the den-

sity gradient at the voxel [13]. ART is much simpler than the Filtered Back Projection method (FBP), which is based on the Fourier transform [14]. For FBP a large number of projections are required for a good reconstruction. In addition, the flow field around a model has to be reconstructed from incomplete projection data due to the obstruction caused by the model. As described by Sourgen et al. [15], obstructed parts have to be filtered or interpolated for the FBP algorithm in order to avoid the numerical artifacts. A comparison between the FBP and ART reconstruction algorithms based on the same LICT (Laser Interferometric Computed Tomography) measurements can be found in Ota et al. [13].

In this paper the new approach to the three-dimensional ART reconstruction and integration process is developed in order to achieve the three-dimensional density distribution based on CBOS data. In previous investigations [13], the Poisson’s equation was used to determine the density distribution [13,14].

The iteration process of the new approach can be described as follows:

where fⁱ denotes the density gradient distribution at step i, P_k stands for the measured displacement vectors in the projection map k, R_kⁱ denotes the estimated displacement vectors from the density distribution f ⁱ at step i, C_k the number of voxels along a projection line; x, y, z represent the coordinates in the density field and X and Y indicate the position in the projection map k.

In order to start the iteration process, the initial distribution of the density gradients f⁰ (x, y, z) is set to zero. Then the displacement vectors are estimated according to Equation (2) for the first projection. Taking into account the measurements for the first projection, a new estimation for the density gradients can be calculated. This process is applied to all projections. The three-dimensional distribution of the density is obtained by a linear integration of the density gradients in the x, y and z directions respectively. At least 30 iterations over the total number of projections are necessary in order to get a converged solution. In order to estimate the accuracy of the reconstruction, several theoretical test cases as well as a comparison with a reconstruction based on the Abel transformation has been carried out [13,14,22]. Earlier studies by Ota et al. [14] showed that good reconstruction data can be obtained from 36 projection angles ranging from 0˚ to 175˚ with intervals of 5˚. This configuration leads to an underestimation for the density of about 7% even in regions with high gradients.

In this case, assuming that the flow is axially symmetrical, every projection perpendicular to the symmetry axis should look the same. Earlier studies by Ota et al. [14] showed that good reconstruction data can be obtained from 36 projection angles ranging from 0˚ to 175˚ with intervals of 5˚. This configuration leads to an underestimation for the density of about 7% even in regions with high gradients. For the reconstruction of the flow field 152 ´ 240 ´ 240 voxels are used, which corresponds to a

distance between the voxels of 0.2 mm.

In Figures 5 to 10 the displacements of the flow around the spike are represented at different instants of time. The size of the interrogation window for the CBOS evaluation is 10 ´ 10 pixels for all images, with an overlapping of 80%. In column a) pseudo color schlieren images are shown. These images show the vertical and horizontal displacements at the same time. In Columns b) and c) the horizontal and vertical displacements are shown respectively. In Column d) the results from the density reconstruction are shown. The color scales for the Figures 5 to 10 can be found in Figure 14.

Figure 5(a) shows the initially expected steady flow pattern, which is known from longer spikes (l/D > 1.2): a shock from the tip, followed by an expansion fan at the end of the conical tip. Furthermore a normal shock at the front of the cylinder can be seen. The isosurface for the density in Figure 5(d) shows the conical shock from the spike tip. The radial distribution of the density in front of the cylinder (Figure 6(d)) indicates the cushion-like high density zone in front of the cylinder. Nevertheless, the density gradients over the conical shock, which is very small compared to the grid size, are already underestimated with the CBOS measurement. Therefore the reconstructed density behind the shock is 12% lower than the value derived from theoretical considerations for conical shocks. In contrast the high density in front of the cylinder is well predicted.