Towards robust BOS measurements for axisymmetric flows

Abstract

Abstract Accurately reconstructing a radial refractive index ( n ) field is challenging in axisymmetric background oriented Schlieren (BOS) measurement. In this study, we systematically investigated several widely adopted inversion algorithms in BOS applications. To quantitatively assess the performance of each algorithm, a synthetic experiment mimicking a helium jet discharged into ambient air was established to provide the reference. Relying on the necessity to solve a Poisson equation for a line-of-sight projected variable, tested algorithms were categorized into two groups: direct and indirect. In the direct approach, the algorithm is applied directly to the light deflection angle ( $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> ) to reconstruct the radial $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> field, defined as $$\delta =(n-n_{0})/n_{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>δ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:mrow> </mml:math> where $$n_{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>n</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> is the reference refractive index. In the indirect group, the Poisson equation is solved first. Then, an inversion algorithm is subsequently applied to the projected $${{\overline{\delta }}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>δ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> to obtain $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> in the radial plane. The two approaches were compared with the synthetic experiment both using the adaptive Fourier–Hankel methods (AFH). The comparison showed that at the cost of introducing the additional step of solving the Poisson equation, the indirect approach performed more accurately when noises were present in the $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> measurements. To identify the proper inversion algorithm suitable for the indirect approach, we further compared four types of algorithms in the synthetic experiments including AFH, onion peeling (OP), three-point Abel (TPA), and filtered back projection tomography (FBPT). The results showed that TPA had the best performance in terms of the reconstruction accuracy with noisy $$\varepsilon $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> data. Finally, experiments on axisymmetric helium jets were conducted to confirm the effectiveness of the proposed TPA algorithm in the indirect approach. Graphic Abstract

Topics & Concepts