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Physically based simulation of focusing schlieren imaging for a hypersonic boundary layer flow

Mingjia Chen, Bing Liu, Fei Qin, Xiao Liu, and Bo Zhou

Open Access Open Access

Abstract

Focusing schlieren systems are more advantageous than conventional schlieren systems in providing a schlieren image with certain spatial discrimination along the light path. The present work employed a hybrid of the optical-transfer matrix and ray-tracing method to faithfully replicate complete physical imaging processes throughout a focusing schlieren optic system. A direct numerical simulation of a hypersonic boundary layer flow was employed to synthesize focusing schlieren images. The influence of various configuration parameters on the properties of focusing schlieren image such as local schlieren structure, brightness, sensitivity, and depth of field were systematically explored. In addition, an approximation method was proposed as a simplified means to facilitate the simulation of a focusing schlieren image.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)
Fig. 1.
Fig. 1. Representative schematic setup of a focusing schlieren imaging system.

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Fig. 2.
Fig. 2. Light propagation (a) in space of constant refraction index with a displacement in the $z$ direction of $r$, (b) through a thin lens, and (c) in the test section with continuously varying refraction index in space. ${{\boldsymbol p}}(x,\;y,z)$ denotes the location on the trace of a light ray with $\varphi$ and $\theta$ being azimuth and elevation angles. In case of ray tracing, ${\boldsymbol {v}}(x,\;y,z)$ is the direction vector of light propagation. The superscript denotes data updated during the iteration of the optical path.

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Fig. 3.
Fig. 3. (a) Conjugated relationship between the source and cutoff grids through a field lens; yellow squares represent bright stripes on the grid. (b) Different relative shift (${D_{\rm s}}$) between the image of the source gird (yellow square) and the cutoff grid that results in bright, dark, and gray schlieren images from left to right, respectively.

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Fig. 4.
Fig. 4. DNS simulation of a hypersonic boundary layer flow for iso-surface density of ${0.3}\;{\rm{kg/}}{{\rm{m}}^3}$. The white arrow indicates the flow direction.

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Fig. 5.
Fig. 5. Focusing schlieren images at plane A in Fig. 4 with varying relative shift ${D_{\rm s}}$ and a LPF of 20 lp/mm. (a) ${D_{\rm s}} = {D_{{\rm LPF}}}$, (b) ${D_{\rm s}} = {0.5}{D_{{\rm LPF}}}$, and (c) ${D_{\rm s}} = {{0}}$ corresponding to bright, gray, and dark schlieren images, respectively.

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Fig. 6.
Fig. 6. Focusing schlieren images with the cutoff grid LPFs of (a) 10, (b) 20, and (c) 40. The bottom panel shows the percentage of light rays within the boundary layer that are deflected to a distance larger than ${D_{{\rm LPF}}}$. The resulting reversal in brightness of schlieren structures is shown in the inserts.

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Fig. 7.
Fig. 7. Focusing schlieren images with focal planes located at (a1–a5) ${{\pm 0}.{5}}\;{\rm{mm}}$ and ${{\pm 1}}\;{\rm{mm}}$ and (c1–c5) ${{\pm 10}}\;{\rm{mm}}$ and ${{\pm 17}.{5}}\;{\rm{mm}}$ apart from (a3 or c3) the central plane along the flow direction, i.e., the plane B in Fig. 4. (b) Spatial correlation coefficients (CC) of the schlieren structures in the red square region as a function of the distance ($d$) from the central plane.

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Fig. 8.
Fig. 8. (a) Correlation of focusing schlieren images (CC) as a function of the distance to the focal plane for different characteristic length scales ${s_h}$. (b) DOFs with $\rm CC = {0.5}$ versus the characteristic length scales of schlieren structures within the boundary layer, which are fitted with a black dashed line.

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Fig. 9.
Fig. 9. Comparison of (a) conventional path-integrated schlieren image, (b) focusing schlieren image simulated based on the ray-tracing method, and (c) its integrated approximation using Eq. (11).

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Tables (1)

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Table 1. Configuration Settings of the Focusing Schlieren System

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Equations (13)

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[ x ′ y ′ φ ′ θ ′ ] = [ 1 0 r 0 0 1 0 r 0 0 1 0 0 0 0 1 ] [ x y φ θ ] ,
[ x ′ y ′ φ ′ θ ′ ] = [ 1 0 0 0 0 1 0 0 − 1 / 1 f f 0 1 0 0 − 1 / 1 f f 0 1 ] [ x y φ θ ] .
n d p d s = v .
d d s ( v ) = ∇ n ,
p i + 1 = p i + Δ s n v i ,
v i + 1 = v i + Δ s ∇ n .
n = 1 + K ρ ,
ε min ∝ D s L L ′ ( L − l ) .
C C = ∑ i = 1 m ( a i − a ¯ ) ( b i − b ¯ ) ∑ i = 1 m ( a i − a ¯ ) 2 ∑ i = 1 m ( b i − b ¯ ) 2 .
R h ( r ) = ⟨ ρ h ′ ( y ) ρ h ′ ( y + r ) ⟩ ,
s h = ∫ 0 ∞ k − 1 E h ( k ) d k ∫ 0 ∞ E h ( k ) d k .
I f ∝ ∫ z G ( σ ) ∗ n y ′ ( z ) d z ,
σ = 0.5 × d A l .
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