^{1}

^{1}

In this study, we apply the optical flow method to the time-series shadowgraph images of impinging jets using a high-speed video camera with high spatial and temporal resolution. This image analysis provides quantitative velocity vector fields in the object space without tracer particles. The analysis results clearly capture the details of the coherent vortex structure and its advection from the shear layer of the free jet. Although the results still leave challenges for the quantitative validation, the results show that this analysis method is effective for understanding the details of the physical phenomenon based on the quantitative values extracted from the shadowgraph images.

Particle Image Velocimetry (PIV) [

In this study, we performed shadowgraph visualizations of impinging jet in transonic flows by applying the algorithm proposed by Liu et al. Numerous studies have revealed the typical flow field of the impinging jet as shown in

The projected-motion equations and physics-based optical flow equation were derived by Liu and Shen [

a fluid medium on an image plane. The perspective center is associated with a camera/lens system. The x_{1}-axis and x_{2}-axis are defined as the image coordinate system. The object coordinates X = (X_{1}, X_{2}, X_{3}) is denoted as the projections of the object-space position vector from the perspective center in this frame.

The projected-motion equation for shadowgraph visualization method is expressed by the following equation as a function of the second-order derivative of the fluid density ρ [

I − I T I T = C ∫ Γ 1 Γ 2 ∇ 12 2 ρ d X 3 . (1)

where, I is the image intensity, I_{T} is the initial image intensity, C is a coefficient related to the setting of the shadowgraph system, and ∇ 12 2 is the Laplace operator expressed by Equation (2):

∇ 12 2 = ∂ 2 ∂ X 1 2 + ∂ 2 ∂ X 2 2 . (2)

The visualized domain is confirmed by two control surfaces, X 3 = Γ 1 = c o n s t . , X 3 = Γ 2 = c o n s t . , corresponding to optical windows. Taking partial differentiation with respect to time in Equation (1) and using the continuity equation expressed by Equation (3), we obtain Equation (4):

∂ ρ ∂ t + ∇ ⋅ ( ρ U ) = 0. (3)

− 1 C ∂ ( I / I T − 1 ) ∂ t = ∇ 12 2 [ ∇ 12 ⋅ ∫ Γ 1 Γ 2 ρ U 12 d X 3 ] . (4)

where, U_{12} = (U_{1}, U_{2}) are the projected two-dimensional velocity components on the coordinate plane (X_{1}, X_{2}) of the fluid.

From Equation (1), the Poisson equation for the integration of the fluid density is obtained:

∇ 12 2 ∫ Γ 1 Γ 2 ρ d X 3 = 1 C ( I / I T − 1 ) . (5)

By putting g = C ∫ Γ 1 Γ 2 ρ d X 3 , the density integral can be obtained by solving the Poisson equation ∇ 12 2 g = I / I T − 1 with appropriate boundary conditions. From Equation (4), we have

∂ g ∂ t + C ⋅ ∇ 12 ⋅ ∫ Γ 1 Γ 2 ρ U 12 d X 3 = 0. (6)

The path-averaged velocity weighted with the fluid density is defined as

〈 U 12 〉 ρ = ∫ Γ 1 Γ 2 ρ U 12 d X 3 ∫ Γ 1 Γ 2 ρ d X 3 . (7)

Substituting Equation (7) into Equation (6), we have

∂ g ∂ t + ∇ 12 ⋅ ( g 〈 U 12 〉 ρ ) = 0. (8)

We note that Horn and Schunck [

To solve the above optical flow problem, a variational formulation with a smoothness constraint has been proposed by Horn and Schunck [

J ( u ) = ∫ A [ ∂ g ∂ t + ∇ ⋅ ( g 〈 U 12 〉 ρ ) ] 2 d x 1 d x 2 + α ∫ A ( | ∇ u 1 | 2 + | ∇ u 2 | 2 ) d x 1 d x 2 . (9)

where α is the Lagrange multiplier. To minimize J(u), by introducing an arbitrary smooth function v = ( v 1 , v 2 ) , computing d J ( 〈 U 12 〉 ρ + p v ) / d p | p = 0 , and using the Green’s theorem where the Neumann condition ∂ 〈 U 12 〉 ρ / ∂ n = 0 is imposed on the domain boundary ∂ A , the Euler-Lagrange equation is given as follows:

g ∇ [ ∂ g ∂ t + ∇ ⋅ ( g 〈 U 12 〉 ρ ) ] + α ∇ 2 〈 U 12 〉 ρ = 0. (10)

To solve Equation (10), Jacobi’s block-wise iteration is used. The solution of Horn & Schunck’s equation is used as an initial approximation for Equation (10) for faster convergence.

^{3}. The compressed air in the tank is released to the atmosphere from the nozzle through a high-pressure pipe, a manual valve, and a plenum chamber. The stagnation pressure in the plenum chamber is measured by a pressure transducer (JTEKT PMS-5M-21M). Atmospheric pressure is measured by a high-precision barometer (GE Druck, DPI740).

The nozzle pressure ratio (NPR) is defined by Equation (13) as the airflow condition:

NPR = P 0 P b . (13)

where P_{0} is the stagnation pressure and P_{b} is the atmospheric pressure.

The NPR was fixed at 1.5, corresponding to the Mach number at the nozzle exit of 0.78 and the flow velocity of 255 m/s. The frame rate was 300 kfps with an exposure time of 1.0 μsec and a bit count of 8 bits. The image resolution was 256 × 128 pixels, equivalent to the actual spatial resolution of 0.14 mm/pixel.

Considering the vertical symmetry of the phenomenon with respect to the mainstream axis, the image measurement area was limited to the upper half area including the nozzle outlet and a part of the wall surface (area enclosed by the dash line in _{t}) between the two images. The d_{t} was set to the minimum time interval of 3.33 μsec.

As shown in

Figures 7-10 show velocity vectors, streamlines, velocity contours, and vorticity contours, respectively, obtained by applying the image analysis using optical flow to the time-series shadowgraph images. Some vectors and streamlines are intentionally decimated for viewability. The velocity contour shown in _{0} = 255 m/s).

^{2}/Hz], and the right vertical axis indicates the PSD [count^{2}/Hz] of the count value of the camera corresponding to the brightness on the image. The peak frequencies of the vortex shedding based on the wall pressure and image analysis are 7162 [Hz] and 6800 [Hz], respectively. There is a good correlation between both, indicating the quantitative validity of the image analysis.

In order to extract quantitative velocity vectors by image analysis without using tracer particles, we applied the optical flow analysis based on the physical relationship between the density change and the brightness change to the time-series shadowgraph images of the impinging jet. We carried out flow visualizations of transonic impinging jet using the high-speed video camera with high spatial and temporal resolution. Since the impinging jet system has a two-dimensional nozzle with a high aspect ratio and side walls, two-dimensionality of the phenomenon is assumed.

We succeeded in capturing the shear layer fluctuations due to KH instability, the detailed spiral structure of the KH vortices, and complicated interference flows near the collision wall by extracting the quantitative velocity vectors with high spatial resolution. The vector fields and streamlines allow us to clearly define the vortex center that is difficult to understand from qualitative shadowgraph images. Besides, we observed that the flow in the shear layer of the free jet is entrained and accelerated by the rotation of the KH vortex. In addition, there was a correlation between the spectral analysis results of both the wall pressure fluctuation and the brightness value fluctuation of the shadowgraph image.

Although there is a still problem in quantitative validation by comparison with PIV measurements and numerical calculations, the analysis results are reasonable compared with previous studies. Our results indicate that this analysis

method is effective for quantitatively extracting velocity fields and understanding flow physics from shadowgraph images.

We would like to sincerely thank T. Liu of Western Michigan University for his many suggestions on this experiment and the analytical method.

The authors declare no conflicts of interest regarding the publication of this paper.

Hijikuro, M. and Anyoji, M. (2020) Application of Optical Flow Analysis to Shadowgraph Images of Impinging Jet. Journal of Flow Control, Measurement & Visualization, 8, 173-187. https://doi.org/10.4236/jfcmv.2020.84011

ρ: Fluid density

I: Image intensity

I_{T}: Initial image intensity

C: Shadowgraph system coefficient

α: Lagrange multiplier

u: Optical flow velocity

U: Local instantaneous velocity

U_{0}: Nozzle outlet velocity

D: Nozzle outlet width

Ω: Vorticity

P_{0}: Stagnation pressure

P_{b}: Atmospheric pressure

X: Object coordinate

x: Horizontal coordinate

y: Vertical coordinate