Extraction of skin friction topology of turbulent wedges on a swept wing in transonic flow from surface temperature images

Abstract

This paper describes a first-order approximate method of extracting the skin friction topology of turbulent wedges from temperature sensitive paint (TSP) images on a NASA Common Research Model with Natural Laminar Flow (CRM-NLF) wing at Mach 0.85 in the National Transonic Facility (NTF) for full-flight Reynolds number testing. The extracted skin friction topology of a typical turbulent wedge exhibits the similarity in the distributions of the surface temperature, skin friction magnitude and skin friction divergence. The developed method computes a normalized skin friction field associated with a surface temperature variation superposed on a simple known base flow. The selection of the relevant parameters in the application of this method is discussed, including the base-flow power-law exponent, relative amplitude of the surface temperature variation to the base flow, Lagrange multiplier, spatial resolution, and filter size.

Graphic abstract

Surface temperature and skin friction fields of a typical turbulent wedge: (a) temperature, (b) skin friction magnitude, and (c) skin friction divergence. The color bar has no unit.

Appendices

Appendix 1

1.1 Parametric Effects

As indicated in Sect. 3.2, the relevant parameters in the proposed approximate method applied to TSP images are the power-law exponent $m$, the relative amplitude $A_{m}$, and the Lagrange multiplier $\alpha$. It is necessary to evaluate the effects of these parameters on extraction of skin friction fields from surface temperature images. To examine this problem, as shown in Fig. 

Fig. 14

A selected region of the surface temperature image for the heating layer case P2523: (left) original image, and (right) image with added random noise. The unit of the color bar is K

14a, a leading-edge region of the surface temperature image for the heating layer case P2523 is selected as a testing field.

Figure 

Fig. 15

Skin friction lines and normalized magnitude fields extracted for different values of the power-law exponent: (left) $m = 0.1$, (middle) $m = 0.3$, and (right) $m = 0.5$, where $A_{m} = - 0.2$ and $\alpha = 10^{ - 3}$. The color bar has no unit

15 shows skin friction lines and magnitude fields extracted from the image in Fig. 14a for $m = 0.1$, $m = 0.3$, and $m = 0.5$, where $A_{m} = - 0.2$, and $\alpha = 10^{ - 3}$. The skin friction topology in these cases remains the same, but the skin friction magnitude fields are different. The effect of $m$ is more pronounced particularly near the leading edge when $m$ is smaller ($m < 0.5$). It is because the base-flow surface temperature gradient $\partial T^{(0)} /\partial x = c_{1} \,x^{2m - 1}$ becomes singular as x approaches to zero. The value of $m$ could be determined by fitting surface temperature data in a suitable base flow. For example, the laminar boundary layer near the leading edge of the swept wing is selected as based flow in the heating layer cases, and the estimated power-law exponent is $m = 0.5$. In many applications, the constant surface temperature gradient $\partial T^{(0)} /\partial x = const.$ could be used a reasonable linearization approximation in a region.

Figure 

Fig. 16

Skin friction lines and normalized magnitude fields extracted for different values of the amplitude: (left) $A_{m} = - 0.3$, (middle) $A_{m} = - 0.6$, and (right) $A_{m} = - 0.9$, where $m = 0.5$ and $\alpha = 10^{ - 3}$. The color bar has no unit

16 shows skin friction lines and magnitude fields extracted for different values of the amplitude:$A_{m} = - 0.3$, $A_{m} = - 0.6$, and $A_{m} = - 0.9$, where $m = 0.5$ and $\alpha = 10^{ - 3}$. The magnitude of $A_{m}$ determines the contribution of the surface temperature variation to skin friction relative to the base flow, which depends on the surface temperature distribution in a region of interest. As shown in Fig. 16, when the magnitude of $A_{m}$ decreases, the skin friction component in the base flow (the main stream component in this case) becomes more visible. Although the sign of $A_{m}$ is determined based on a physical argument (see Sect. 3.2), the magnitude of $A_{m}$ as a free parameter is not known a priori. It could be determined in situ based on reliable CFD and other global skin friction measurement methods such as oil film techniques. In this work, the magnitude of $A_{m}$ is not calibrated.

Figure 

Fig. 17

Skin friction lines and normalized magnitude fields extracted for different values of the Lagrange multiplier: (left) $\alpha = 10^{ - 4}$, (middle) $\alpha = 10^{ - 3}$, and (right) $\alpha = 10^{ - 2}$, where $m = 0.5$ and $A_{m} = - 0.2$. The color bar has no unit

17 shows skin friction lines and magnitude fields extracted for different values of the Lagrange multiplier: $\alpha = 10^{ - 4}$, $\alpha = 10^{ - 3}$, and $\alpha = 10^{ - 2}$, indicating the effect of the Lagrange multiplier on the extracted results, where $m = 0.5$ and $A_{m} = - 0.2$. In the Euler–Lagrange equation Eq. (5), the Lagrange multiplier $\alpha$ acts as a diffusion coefficient. For a larger value of $\alpha$, the solution for a skin friction field is smoother, while when $\alpha$ is small, the solution tends to capture finer structures. According to the error analysis, when the surface temperature gradient magnitude $\left\| {\nabla T} \right\|$ is small, the Lagrange multiplier $\alpha$ must be sufficiently small to reduce the error in skin friction. However, the solution is more sensitive to the data error as $\alpha$ decreases. Therefore, there is an optimal value of $\alpha$ to meet the two conflicting requirements. As shown in Fig. 17, $\alpha = 10^{ - 3}$ is a suitable value in this case.

In addition, the effects of the spatial resolution and image noise are evaluated. Figure 

Fig. 18

Skin friction lines and normalized magnitude fields extracted temperature images with different spatial resolutions: (left) 152 × 100, (middle) 380 × 250, and (right) 760 × 500 pixels, where $m = 0.5$, $A_{m} = - 0.2$ and $\alpha = 10^{ - 3}$. The color bar has no unit

18 shows skin friction lines and magnitude fields extracted for different spatial resolutions: 152 × 100, 380 × 250, and 760 × 500 pixels, where $m = 0.5$, $A_{m} = - 0.2$ and $\alpha = 10^{ - 3}$. As the spatial resolution is decreased, the extracted skin friction structures are degraded particularly in the laminar flow regions between turbulent wedges. When the spatial resolution is lower than a critical value (such as 152 × 100 pixels in this case), the turbulent wedge structures cannot be correctly extracted.

The image noise should be somewhat removed using a filter to obtain an acceptable solution of a skin friction field. To evaluate the effect of noise, the image with added random noise in Fig. 14b is processed. Figure 

Fig. 19

Skin friction lines and normalized magnitude fields extracted from noisy images filtered using the Gaussian filter with different standard deviations: (left) 1, (middle) 5, and (right) 10 pixels, where $m = 0.5$, $A_{m} = - 0.2$ and $\alpha = 10^{ - 3}$. The color bar has no unit

19 shows skin friction lines and magnitude fields extracted from noisy images filtered using the Gaussian filter with different standard deviations: 1, 5, and 10 pixels, where $m = 0.5$, $A_{m} = - 0.2$ and $\alpha = 10^{ - 3}$. For noisy TSP images, filtering is required to obtain reasonable results.

Appendix 2

2.1 Transition front

The current CFD (the e^N-mehod) can give a natural transition front on the CRM wing, but cannot predict the forced transition by roughness (Lynde et al. 2019). A question is how to find the natural transition front in the boundary layer contaminated by many turbulent wedges to compare with the CFD result. As shown in TSP image in Fig. 2a, turbulent wedges triggered by imperfect surface could make identification of the natural transition front ambiguous and difficult. Further, a relevant question is how to determine the length of a turbulent wedge or the downstream endpoint of a turbulent wedge particularly when neighboring turbulent wedges merge. Based on observations of skin friction lines superposed on the skin friction divergence field, the downstream endpoint of the positive skin friction divergence region corresponds to the endpoint of the attachment line associated with a turbulent wedge. Here, an endpoint of the positive skin friction divergence is defined as a point across which the divergence rapidly decreases to zero. An endpoint of an attachment line is defined as a point across which skin friction lines are no longer diverge. Therefore, a line connecting the endpoints of the positive skin friction divergence regions associated with turbulent wedges could be approximately considered as the transition front in the mixed (contaminated) region on the swept wing. To obtain a complete transition front, an interpolating scheme can be developed to combine the endpoints identified in skin friction divergence map and the natural transition front identified in surface temperature map.

Figure 

Fig. 20

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a Transition front identified as a line connecting the endpoints of the positive skin friction divergence regions associated with turbulent wedges for the case P2523, and b transition front identified from TSP image and predicted by Lynde et al. (2019). The material in b is declared a work of the U.S. Government and is not subject to

20a shows the transition front identified as a line simply connecting the endpoints of the positive skin friction divergence region associated with turbulent wedges for the case P2523. As shown in in Fig. 20b, Lynde et al. (2019) estimated the transition front by connecting the endpoints of visible turbulent wedges in the TSP image and found reasonable agreement between the observed transition front and CFD transition predictions for the clean laminar boundary layer. In Fig. 20b, the observed endpoints of turbulent wedges near the wing root are located more downstream than those of the positive skin friction divergence regions in Fig. 20. However, the boundary layer near the wing root is highly contaminated by the junction vortex and local 3D separation such that it is no longer a clean laminar boundary layer. It is still questionable how to identify the transition from this contaminated boundary layer. The skin friction divergence field provides a more rational indicator for turbulent wedges, which has the clear physical and topological meanings.