Global skin friction measurements and interpretation

Progress in Aerospace Sciences xxx (xxxx) xxx

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Global skin friction measurements and interpretation Tianshu Liu Department of Mechanical and Aerospace Engineering, Western Michigan University, Kalamazoo, MI, 49008, USA

A R T I C L E I N F O

A B S T R A C T

Keywords: Skin friction Topology Surface flow visualization Image-based measurement Luminescent oil Surface temperature Surface pressure Surface scalar Complex flows Optical flow Variational method Inverse problem

From a unified perspective, this review describes global skin friction diagnostics based on surface flow visuali­ zations. The foundations of the developed image-based methods are relations between skin friction and other measurable surface quantities (e.g. oil-film thickness, temperature, scalar density and pressure), which are derived from the relevant governing equations in fluid mechanics. Interestingly, these relations can be re-cast into a generic form of the optical flow equation in the image plane, and therefore skin friction fields can be extracted from surface visualization images by using a variational method as an optical flow problem. Global skin friction diagnostics in typical experiments are described, including experimental setups, image processing, and topological interpretation of extracted skin friction fields.

1. Introduction 1.1. Skin friction topology and near-wall flow Skin friction is a wall shear stress vector in viscous flow, which is one of the fundamental surface quantities in fluid mechanics. The conven­ tional notation for skin friction is τ w , and for simplicity skin friction is denoted by τ in this paper. Typically, a skin friction field (τ -field) in a three-dimensional (3D) complex flow contains distinct topological fea­ tures such as critical points and separation and attachment lines. Skin friction topology reveals near-wall structures of complex flows. To observe skin friction topology in experiments, surface flow visualiza­ tions have been conducted by using oil mixed with particles on a surface. From streaky oil patterns on the surface, topological structures are conjectured based on observer’s intuition and experience. However, this approach is not only somewhat subjective, but also unable to identify some subtle features particularly attachment lines. In general, it is difficult to unambiguously infer topological structures in skin friction fields by observing surface flow visualizations without quantitative image processing and analysis. To resolve this problem, results in the critical-point theory of differential dynamical systems, differential ge­ ometry and surface topology can be adopted for a rational description of skin friction topology [1–6]. Topological methods were introduced by Legendre [7] and Lighthill

[8] to interpret surface oil visualization patterns in 3D separated flows. In particular, Lighthill [8] indicated that the Poincare-Hopf index the­ orem for a vector field on a closed surface could be used as a topological constraint on a skin friction field. The application of the Poincare-Hopf index theorem was extended by Hunt et al. [9] for several flows, and particularly a useful topological rule was proposed for a skin friction field in junction flow. The topological rules derived from the Poincare-Hopf index theorem provide necessary constraints to identify some hidden critical points that cannot be observed in visualizations. A generalized form of the Poincare-Hopf index theorem was given by Ma and Wang [10] for a vector field on a manifold with a non-penetrable boundary, which led to a topological rule similar to that given by Hunt et al. [9]. Surface flow visualizations are conducted on a limited region enclosed a penetrable boundary defined as a line which vectors can freely penetrate through. To deal with a penetrable boundary for extended applications, Foss [11,12] applied the Poincare-Hopf index theorem to a collapsed sphere with the lateral edges called seams where vectors are required to be tangential to the boundary, and considered inflow and outflow across the boundary through holes on the sphere. Now, the topological notations have been widely used to describe structures of complex separated flows [14–25]. Further, considering a penetrable boundary in a skin friction field, Liu et al. [13] applied the Poincare-Bendixson index formula (the P-B formula in short) to various cases, which presents a conservation law

E-mail address: [email protected]. https://doi.org/10.1016/j.paerosci.2019.100584 Received 7 September 2019; Received in revised form 16 October 2019; Accepted 16 October 2019 0376-0421/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Tianshu Liu, Progress in Aerospace Sciences, https://doi.org/10.1016/j.paerosci.2019.100584

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between isolated critical points and boundary switch points. In simple notations, the P-B formula is expressed as #N

#S ¼ 1 þ ð #Z þ

#Z Þ=2;

by a skin friction field and a surface pressure field. From this perspective, skin friction topology is a key to understand near-wall structures in complex flows [27].

(1)

1.2. Global skin friction diagnostics

where #N and #S denote the numbers of nodes and saddles, respec­ tively, and #Zþ and #Z denote the numbers of positive and negative switch points, respectively. If there is a point Z on a boundary at which the neighboring inflow and outflow segments are divided, this point is called a switch point. There are two types of switch points. Fig. 1 il­ lustrates nodes, saddles, and negative and positive switch points on a penetrable boundary with inflow and outflow segments. By following a skin friction line (or a streamline), if vector in a sufficiently small neighborhood of Z moves inward first and then outward across a boundary, this switch point is negative, which is denoted by Z . Otherwise, a switch point is positive, which is denoted by Zþ , if vector moves outward first and then inward across a boundary in a neighbor­ hood of Zþ . When inflow and outflow across a closed boundary remain qualitatively invariant, i.e., #Zþ #Z is fixed, Eq. (1) indicates that nodes and saddles enclosed by the boundary must occur or disappear in pairs. The classical results, such as the hairy sphere theorem [8], the topological rule given by Hunt et al. [9] for junction flows, and Foss’ collapsed sphere method [11], can be derived from Eq. (1). In fluid mechanics, a focus or a spiraling node is often observed [24], at which neighboring skin friction lines (streamlines) spiral either inwardly or outwardly. Foci in a skin friction field are usually associated with tornado-like vortices (see Example 1). In topology, a focus is still counted as a node in spite of its physical meaning. The significance of skin friction is that it is one of the two elemental constitutes (along with surface pressure gradient) to completely recon­ struct near-wall flow structures. This fact is established by the Taylorseries expansion solution of the Navier-Stokes (NS) equations for an incompressible viscous flow [26]. The non-dimensional velocity us normalized by a reference velocity uref parallel to the surface is given by the Taylor-series expansion in the non-dimensional wall-normal coor­ dinate x3 , i.e., � � � x2 x3 (2a) us ¼ Rel x3 τ þ 3 rp þ 3 ðLτ rðr ⋅ τ ÞÞ þ O x43 ; 2 6

Global skin friction diagnostics give high-resolution skin friction fields extracted from certain measurable quantities in visualizations. Several global direct or semi-direct techniques were developed based on relationships between shear-induced deformation of special material structures and measurable quantities, including shear-sensitive liquid crystals [28–30], surface-stress-sensitive polymer film [31,32] and micro-pillar arrays [33–35]. Global skin friction diagnostics discussed in this paper are based on conventional surface flow visualizations in experimental fluid mechanics, including surface oil film, heat transfer, mass transfer, and pressure visualizations. The main measured quanti­ ties are oil-film thickness, surface temperature, surface scalar concen­ tration, and surface pressure. To extract skin friction from these quantities is an inverse problem. Relations between skin friction and measured quantities are derived by projecting the relevant governing equations for surface flow visualizations onto the image plane. This projection method was originally used to derive the physics-based op­ tical flow equation for global velocity diagnostics from various flow visualizations [36]. For luminescent oil-film visualization on a surface, an equation relating the projected skin friction to the normalized luminescent in­ tensity in the image plane was derived from the thin-oil-film equation [37]. Interestingly, this equation has the same mathematical form as the physics-based optical flow equation. Therefore, extracting skin friction from luminescent oil images can be solved as an optical flow problem by a variational method. This technique called a global luminescent oil-film (GLOF) skin friction meter has been used to measure skin friction fields in complex separated flows [38–47]. In addition, oil mixed with lumi­ nescent particles was used to obtain particle velocity field that is pro­ portional to skin friction field by cross-correlation method [48,49]. Further, the projection method was applied to the energy and mass transport equations for surface heat transfer and mass transfer visuali­ zations with coating techniques (temperature and pressure sensitive paints and sublimating coatings), and relations between skin friction and surface temperature and scalar concentration were obtained [50–54]. Recently, a relation between skin friction and surface pressure was derived from the Navier-Stokes equations [55]. These relations can be re-cast into a special form of the optical flow equation in the image plane. Therefore, skin friction fields can be sought as an inverse problem from surface temperature, scalar concentration and pressure images. This paper provides a unified framework for global skin friction di­ agnostics based on surface flow visualizations and interpretation of skin friction fields. The focus is placed on the fundamental relations between skin friction and measured quantities (e.g. oil-film thickness, surface temperature, scalar concentration and pressure) and the physical meanings of the relevant terms. These relations can be re-cast to the optical flow equation in the image plane, and thus a variational method can be applied to extraction of skin friction fields from surface visuali­ zation images. Global skin friction diagnostics using the developed methods in various complex flows are described as examples.

where τ ¼ τ =ρ u2ref is the non-dimensional skin friction, p ¼ p= ρ u2ref is

the non-dimensional surface pressure, where uref is a reference velocity, xi (i ¼ 1; 2; 3) are the coordinates normalized by a length scale l, r ¼ ∂= ∂xk (k ¼ 1; 2) is the non-dimensional gradient operator on the surface, L ¼ ðl2 =νÞ∂=∂t r2 is a non-dimensional differential operator, Rel ¼ l uref =ν is the Reynolds number, and ν is the kinematic viscosity of fluid. The non-dimensional vertical (wall-normal) velocity component is � � � x23 x33 2 (2b) u3 ¼ Rel r⋅τ r p þ O x43 ; 2 6 According to Eq. (2), a near-wall velocity field is solely determined

2. Skin friction from luminescent oil-film visualization 2.1. Skin friction and oil-film development Quantitative skin friction measurements were made by using imagebased interferometry for determining the thickness of an oil drop on a surface [56]. Data processing in interferometric oil-film skin friction meter is based on a local similarity solution of the thin-oil-film equation providing relationship between skin friction and oil-film thickness change. However, this similarity solution is no longer applicable to

Fig. 1. Illustration of nodes, saddles, and positive and negative switch points. 2

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global measurements of a continuous oil film in a large region on a surface. To extract high-resolution skin friction field from oil-film thickness measurement, the thin-oil-film equation should be solved globally as an inverse problem. Here, there are two related problems: (1) how to optically measure continuous oil-film thickness distribution, and (2) how to extract a skin friction field from measured time-dependent oil-film thickness distribution. Global luminescent oil-film (GLOF) skin friction meter was devel­ oped by Liu et al. [37]. Luminescent oil is used since the oil-film thickness is proportional to the luminescent emission intensity when the oil is optically thin. In experiment, a thin luminescent oil film is brushed or sprayed on a surface in a region of interest before starting a wind tunnel, and it is illuminated by light sources with a suitable wavelength such as ultraviolet (UV) lights. Therefore, oil-film thickness measurement is converted to luminescent intensity measurement by using a camera with a suitable optical filter. After flow is turned on, a time sequence of evolving GLOF images is acquired. For an optically thin luminescent oil film applied to a surface under suitable illumination, the oil-film thickness h is proportional to the normalized luminescent intensity, i.e., h ¼ β 1 ðI =Iex Þ, where I is the luminescent intensity, Iex is the intensity of the illumination light on the surface, and β is a coefficient proportional to the quantum efficiency of luminescent dye seeded in the oil. We consider an orthographic pro­ jection transformation between the surface coordinates ðX1 ; X2 Þ in the object space and the image coordinates ðx1 ;x2 Þ, i.e., ∂= ∂Xi ¼ λ ∂= ∂ xi , where λ is a scaling constant that is approximately a ratio between the focal length of a camera/lens system and its distance to the surface. Define the normalized luminescent intensity gI ¼ I=Iex as a measurable quantity that eliminates the effect of non-uniform illumination. Replacing h by I in the thin-oil-film equation and using the ortho­ graphic projection transformation, we obtain the following equation

∂ gI τ Þ ¼ fI ; þ r⋅ðgI b ∂t

� gI r

∂ gI τÞ þ r ⋅ ðgI b ∂t

� fI þ α r2 b τ ¼ 0;

(6)

where r2 ¼ ∂2 =∂ xi ∂ xi (i ¼ 1; 2) is the Laplace operator. The Neumann condition ∂b τ =∂n ¼ 0 is imposed on an image domain boundary ∂D. The standard finite difference method is used to solve Eq. (6) with the Neumann condition. The numerical solution of Eq. (5) gives the equiv­ alent skin friction b τ and thus the relative projected skin friction vector bτ = g is obtained. The Lagrange multiplier plays a role in controlling the diffusion term in Eq. (6). As a rule of thumb, a sufficiently small value of α is selected to preserve sharp features as long as a numerical solution of Eq. (6) converges. Selection of a suitable value of α can be made based on simulations on synthetic images. When GLOF images contain rich features with sufficiently large intensity gradient, a solution of Eq. (6) is not sensitive to the Lagrange multiplier in a considerable range of α. Development of a luminescent oil film on a surface is time-dependent even in steady flow. The numerical solution of Eq. (6) gives a snapshot skin friction field from a pair of successive images. From a time sequence of GLOF images, a series of snapshot solutions at successive moments are obtained. Physically speaking, a snapshot solution captures salient skin friction signatures in regions where the oil-film evolution is more sen­ sitive to flow at that moment. Therefore, a time sequence of snapshot solutions is required to capture major skin friction signatures at different moments during the oil-film evolution process. To reconstruct a relative steady-state or time-averaged skin friction field, snapshot solutions are superposed or averaged. To improve the reconstruction accuracy, an adaptive averaging procedure could be used by truncating the number of snapshot solutions for averaging in different regions based on a history of the local oil-film evolution [42]. In general, by using the GLOF method, a normalized skin friction field is given without in-situ cali­ bration. An absolute skin friction field can be obtained by in-situ cali­ bration that utilizes some accurate values of skin friction at several reference locations (at least one) given by reliable techniques like an interferometric oil-film skin friction meter or computational and theo­ retical methods. In an error analysis, the image intensity and skin friction vector are decomposed into a basic solution and an error, i.e., gI ¼ gI0 þ δ gI and b τ ¼ bτ 0 þ δ bτ , where gI0 and bτ 0 satisfy exactly Eq. (6), and δ bτ is the resulting error in skin friction and δ g is an error in the luminescence measurements. Substitution of the above decompositions into Eq. (6) leads to an error propagation equation and then a formal estimate of the relative error [49]. � � �� δ t gI; tt ðδb τ ÞN α ðδb τ ÞN r 1 r2 ¼ ; (7) τ 0 k 2 krgI0 k char kbτ 0 k gI0 krgI0 kchar τ 0k kb kb

(3)

where b τ ¼ τ gI ðλ =2μo βÞ is an equivalent skin friction vector, and τ ¼ ðτ1 ; τ2 Þ is a skin friction vector projected onto the image plane ðx1 ; x2 Þ, and r ¼ ∂=∂ xi (i ¼ 1; 2) is the gradient operator. The right-hand-side (RHS) term in Eq. (3) is defined as � � g3I fI ¼ λr⋅ ðλrp ρo aÞ ; (4) 2 3μo β where μo is the oil viscosity, ρo is the oil density, and a is the gravity vector. The term fI represents the effects of the pressure gradient and gravity, which is in the order of λh3 << 0 since λ << 0 and h3 < < 0. Therefore, this term is considered as a higher-order small term in Eq. (3). Interestingly, Eq. (3) has the same form of the physics-based optical flow equation for various flow visualizations [36].

where ðδb τ ÞN ¼ δbτ ⋅NT is the projected skin friction error along the unit � � normal vector NT ¼ rgI0 =�rgI0 �, gI;tt ¼ ∂2 gI0 =∂t2 is the second-order time derivative, δ t is a time step between two consecutive images, � � �rgI0 � is a characteristic intensity gradient magnitude, and kb τ 0 k is a char characteristic value of skin friction (e.g. the mean value). The symbol r 1 in Eq. (7) is a symbolic inverse operator for the solution of the partial differential equation rφ ¼ b. The first term in the RHS in Eq. (7) represents an elemental error in � � the time differentiation. Since δ t and �rgI0 �char are finite, a constraint is � � 1 δ t �rgI0 �char � d, where d is a small positive constant, Therefore, a finite skin friction error δ b τ always exists, which imposes an ultimate limita­ tion on the accuracy of extracting skin friction from GLOF images. For a given time step δ t, larger intensity gradient leads to a smaller skin friction error. Rich local features with large intensity gradient in GLOF images ensure the accuracy in computations. The second term in the RHS in Eq. (7) represents the effect of the Lagrange multiplier, which is � � 1 � � proportional to is α�rgI0 � . Therefore, when �rgI0 � is small, the

2.2. Variational formulation Solving the single equation, Eq. (3), for the two unknown compo­ nents b τ ¼ ðbτ 1 ; bτ 2 Þ is a typical inverse problem, in which additional constraint is required for a unique solution. Therefore, a variational method is used for solving this problem. We consider a functional with a smoothness regularization term on an image domain D, which is defined as �2 Z �� Z � �2 � �2 � ∂ gI � � � � Jðb τÞ ¼ τ Þ fI dx1 dx2 þ α þ r⋅ð gI b �rbτ 1 � þ �rbτ 2 � dx1 dx2 ; ∂t D D (5) where α is a Lagrange multiplier. Minimizing Jðb τ Þ leads to the EulerLagrange equations, i.e.,

char

char

Lagrange multiplier α must be small to reduce the error.

3

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In addition to the variational method, an alternative is to extract a skin friction field from a sequence of GLOF images by using a leastsquares method [45]. For sufficient number of image pairs, finite dif­ ference forms of the optical flow at different locations and times can be re-arranged into an over-determined system of algebraic equations for skin friction as a least-squares problem. In this approach, the averaging procedure in the time domain is incorporated in a least-squares solution. 2.3. Example 1: Low-aspect-ratio rectangular wing Skin friction topology on a low-aspect-ratio rectangular wing at high angles of attack (AoA) was studied by using the GLOF method in a lowspeed wind tunnel with a test section of 0.406 by 0.406 m at the Applied Aerodynamics Laboratory of Western Michigan University [13]. A glass window at the top of the test section allowed optical access. The free­ stream turbulent intensity was about 0.2%. The temperature sensitivity of the luminescent material was not a concern for low-speed flows with a constant temperature. Fig. 2 shows schematic of a setup for this exper­ iment. This wing had a NACA0012 airfoil section and the wing aspect ratio (AR) of 1.2. The wing chord and span were 127 mm and 152 mm, respectively. The plastic wing was coated by a white LustreKote paint to enhance the luminescent emission detected by a camera. Before a run, luminescent oil (silicone oil mixed with oil-based luminescent dye) was brushed on the wing surface. Oil was excited by two UV lights that were arranged to ensure a uniform illumination field in the test area. A 550 nm long-pass filter was used to filter the light captured by the camera allowing only detection of the luminescent oil emission centered at approximately 590 nm. The wind tunnel was run in a dark environ­ ment, and images were captured by an ISG Lightwise CMOS camera by using Streampix 4.0 image acquisition software (NorPix) at 25 frames per second. The acquired images were then processed by using a specialized Matlab code to extract skin friction fields. The freestream velocity was 20 m/s and the Reynolds number based on the chord was 1.72 � 105. Fig. 3 shows typical GLOF image, skin friction vectors and skin friction lines on the upper surface of the low-AR rectangular wing at α ¼ 18o . The skin friction field extracted by solving Eq. (6) on the upper surface has 193 � 251 vectors (193 vectors in the vertical axis and 251 vectors in the horizontal axis). Skin friction lines indicate complicated topological structures in this separated flow. Separation line starts at the saddle S1 near the leading edge, extending from S1 to the wing tips along the spanwise direction. The primary spiraling nodes (foci) N3 and N4 are connected to S2 and S3 , respectively. Two small spiraling nodes (foci) N1 and N2 occur near the wing tips, which are also connected to the saddles S2 and S3 , respectively. Other two saddles S4 and S5 occur below N3 and

Fig. 3. GLOF measurement on the upper surface of a low-AR rectangular wing at AoA ¼ 18� , (a) typical luminescent oil image, (b) skin friction vectors, and (c) skin friction lines. From Liu et al. [13].

N4 . In the extracted skin friction field on the upper surface, there are 4 nodes and 5 saddles. Separation and attachment lines between the isolated critical points can be clearly identified. From a phenomenological perspective, a sep­ aration line can be defined as a skin friction line to which all neighboring skin friction lines converge asymptotically. Similarly, an attachment line is a skin friction line from which all neighboring skin friction lines

Fig. 2. Illustration of luminescent oil-film skin friction measurement setup. From Liu et al. [13]. 4

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diverge. The typical separation patterns from a saddle to a spiraling node can be found in Fig. 3(c), such as the combinations of S2 N1 , S3 N2 , S4 N3 , and S5 N4 [21–24]. Since a spiraling node (focus) is associ­ ated with a tornado-like vortex [17], skin friction topology on the upper surface results from the interaction between the tornado-like vortices and the tip vortices. A question is whether this extracted topology is correct. To examine this problem, the P-B formula, Eq. (1), is applied. As shown in Fig. 3(c), a closed boundary that consists of the leading, side and trailing edges of the wing (a polygon ABCDA) is considered. Isolated critical points enclosed by the boundary and switch points on the boundary are iden­ tified. Inflow and outflow segments near the leading edge are separated by negative switch points Z1 and Z2 . Two positive switch points Z1þ and Z2þ on the wing tips are mainly induced by the spiraling nodes (foci) N1 and N2 . Negative switch points Z3 and Z4 near the wing tips are induced by the interaction between the tip vortices and the tornado-like vortices associated with the primary spiraling nodes (foci) N3 and N4 . Near the trailing edge, negative switch points Z5 and Z6 are located at the corners where inflow and outflow segments are separated. In Fig. 3 (c), there are 4 nodes and 5 saddles, 6 negative switch points and 2 positive switch points, such that #N #S ¼ 1 þ ð#Zþ #Z Þ=2 ¼ 1. Therefore, the P-B theorem is satisfied in the extracted skin friction field on the upper surface of the wing. When the flow was tripped by a 75 μm thick roughness strip along the leading edge, a dramatic topological change was observed. Skin friction lines on the upper surface in the tripped flow for α ¼ 18o are shown in Fig. 4. Compared to the non-tripped flow in Fig. 3(c), the to­ pological structure near the leading edge is significantly altered although the two primary spiraling nodes are approximately located at the same locations. Skin friction lines converge along the leading edge, indicating leading-edge separation. The flow is partially re-attached at N1 , and then the secondary separation takes place at S1 . Positive switch point Z1þ and negative switch point Z1 occur along the left side of the leading edge, while Z2þ and Z2 occur on the right side. Negative switch points Z3 and Z4 on the side edges are clearly associated with the saddles S2 and S3 . Interactions between saddles S4 and S5 and spiraling nodes (foci) N2 and N3 are responsible to the generation of negative

switch points Z5 , Z6 , Z7 and Z8 . In this case where #N ¼ 3, #S ¼ 5, #Zþ ¼ 2 and #Z ¼ 8, the P-B theorem is also satisfied. 2.4. Example 2: Wing-body junction Skin friction fields in a wing-body junction flow were extracted from GLOF images taken at different viewing angles and positions [44]. The experiments were conducted in a low-speed recirculation wind tunnel in Shenyang Aerospace University. The test section of the tunnel is 3 m long, 1.2 m wide and 1 m high. The flow speed range is 2–55 m/s, and the freestream turbulence intensity is below 0.2%. A “Rood” wing was used, which was a combination of a 3:2 elliptic nose with a NACA0020 tail section attached at its maximum thickness. Fig. 5 shows the Rood wing model vertically mounted on a flat aluminum plate to form a wing-body junction. The wing chord (c), maximum thickness (T) and height (H) are 383.3 mm, 90.0 mm and 235.1 mm, respectively. The length, width and thickness of the aluminum plate with a flat-headed leading edge were 1.25 m, 0.5 m and 9 mm, respectively. The plate that was parallel to the wind-tunnel floor was mounted on four 70-mm-high struts. The leading edge of the wing was at x ¼ 450 mm from the leading edge of the flat plate. The angle of attack (AoA) of the wing could be adjusted. As shown in Fig. 5, the wing surface and neighboring region on the plate were painted with a white prime coating to enhance the emission from the luminescent oil. The luminescent oil was prepared by mixing a commercial ultraviolet (UV) dye with the silicon oil (200-cs). A thin oil film (about 20–40 μm thick) was coated on the model surface using an air spray gun. The parameter used to characterize the wing-junction is the blunt­ ness factor defined as BF ¼ 0:5ðR0 =XT Þ ðT =ST þ ST =XT Þ, where R0 is the leading-edge radius of the wing, XT is the chordwise location of the maximum thickness of the wing, T is the maximum thickness, and ST is the distance from the leading edge along the airfoil surface to the maximum thickness. The Rood wing used in this work has BF ¼ 0.32. The tests were run at the freestream velocity of 27 m/s for AoAs of 0� , 6� and 12� . The Reynolds number based on the maximum wing thickness was ReT ¼ 1:51 � 105 (corresponding to Rec ¼ 6:4 � 105 ). The boundary layer on the flat plate without the Rood wing was surveyed by using a single hot-wire sensor at x ¼ 388, 402 and 416 mm, where x is the streamwise coordinate from the flat-plate leading edge These measure­ ment positions are marked as P1, P2 and P3 in Fig. 5, respectively. Ve­ locity profile measurements indicated that the boundary layer was fullydeveloped turbulent. The tunnel had optical access only in the front and top windows of the test section, imposing limits on setting up multiple cameras and lights to view simultaneously the whole surface of the wing-body junction model. Therefore, to obtain skin friction fields on the whole surface, a single camera was used to acquire GLOF images from different viewing angles and locations in repeated runs. Four 12W UV light

Fig. 4. Skin friction lines on the upper surface of a low-AR rectangular wing at AoA ¼ 18� where the flow is tripped by a roughness strip at the leading edge. From Liu et al. [13].

Fig. 5. The Rood wing vertically mounted on the flat aluminum plate in the test section. From Zhong et al. [44]. 5

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emission device (LED) lamps with the peak wavelength of 365 nm were used to illuminate the oil film. The luminescent emission of the oil was at the wavelength of 550–600 nm. An 8-bit CCD camera was used to record images of 1024 � 1024 pixels, and a 570-nm long-pass filter was placed on the lens of the CCD camera to cut off the illumination light. The camera and the lamps were placed at several different locations for the better illumination and viewing angles of the camera on the wing-body junction model surface. For imaging, the model surface (the wing/floor surface) was approximately divided into five portions: wingtip surface, wing suction surface, wing pressure surface, floor suction surface and floor pressure surface. A time sequence of GLOF images of a specific portion was acquired by using the camera at a given position in one run, and then images of other portions were obtained by using the camera at different positions in repeated runs until the entire surface was imaged. Fig. 6 shows typical GLOF images taken from the five different viewing angles and positions for AoA ¼ 6� on floor in the suction side, floor in the pressure side, wingtip (or top) surface, pressure surface, and suction surface of the wing. The camera was calibrated/oriented at different viewing angles and positions based on several targets (markers) and feature points (at least 6 points) with the known coordinates on the model surface in the body coordinate system. Thus, GLOF images taken from different viewing angles and positions were mapped onto the surface meshes by using the perspective projection transformation (the photogrammetric collin­ earity equations) to reconstruct GLOF intensity fields on the surface. Fig. 7 shows a typical GLOF intensity field mapped on the surface mesh

from five raw GLOF images taken from different viewing angles for AoA ¼ 6� . Skin friction vectors in the image plane were extracted from GLOF images taken from different viewing angles and positions by solving Eq. (6). Fig. 8 shows skin friction lines extracted from GLOF images taken

Fig. 7. GLOF intensity field mapped on the surface mesh from five raw images taken from different viewing angles for AoA ¼ 6� . From Zhong et al. [44].

Fig. 6. Typical GLOF images taken from the five different viewing angles and positions for AoA ¼ 6� : (1) floor in the suction side, (2) floor in the pressure side, (3) wingtip (or top) surface, (4) pressure surface, and (5) suction surface of the wing. From Zhong et al. [44]. 6

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from different viewing angles and positions (see Fig. 6) for AoA ¼ 6� . Then, a complete skin friction vector field on the surface mesh in the 3D object space was reconstructed by using the perspective projection transformation from the extracted skin-friction vectors in the image plane. Fig. 9 shows skin friction lines mapped on the surface of the wingbody junction viewed from the suction and pressure sides of the wing for AoA ¼ 6� , where the color map shows the luminescent intensity distri­ bution of the oil film. The primary separation line originating from the saddle S1 and extending around the wing are associated with the pri­ mary horseshoe vortex on the floor. The node N1 is at the leading edge of the wing intersecting with the floor. Attachment lines and secondary separation lines are observed on the floor near both the suction and pressure surfaces of the wing. On the wingtip surface, a node N2 near the leading edge, a node N3 near the side, and a saddle S2 near the centerline are observed along with separation and attachment lines. On the suction surface of the wing, a distinct saddle S3 exists near the upper corner of the trailing edge, and a long separation bubble stretched in the spanwise direction occurs near the location of maximum thickness of the wing. On

the pressure surface of the wing, a shorter separation bubble occurs. Fig. 10(a) shows a zoomed-in view of the front region of the wingtip surface of the wing, where a sink node N2 , a source node N3 and a saddle S2 are clearly observed. In addition, on the boundary, two negative þ switch points (Z1 and Z2 ) and two positive switch points (Zþ 1 and Z2 ) are identified. Therefore, the P-B formula holds, verifying the topological consistency of the extracted skin friction field in the region. Fig. 10(b) shows a zoom-in view of a long separation bubble on the suction side of the wing, where a single saddle S3 near the upper corner and 5 negative switch points and one positive switch point are identified. The P-B for­ mula is also satisfied in this region. Similarly, as shown in Fig. 10(c), the P-B formula holds in the region of a shorter separation bubble on the pressure surface of the wing.

Fig. 8. Skin-friction lines extracted from GLOF images taken from different viewing angles and positions (see Fig. 6) for AoA ¼ 6� . From Zhong et al. [44]. 7

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Relation between skin friction and surface temperature can be derived from the energy equation of flow [50]. For a flat surface where effect of the surface curvature is zero, this on-wall relation is written as

τ ⋅ rTw ¼ μfQ ; where the virtual source term on a surface is given by � � � 3 � � � 1 ∂ μ ∂Φ ∂T fQ ¼ þa ; ar2 qw þ ρc ∂x3 w k ∂t ∂ðx3 Þ3 w

(8)

(9)

Tw is the surface temperature, ρ, c, μ and a ¼ k=ρc are the density, spe­ cific heat, dynamic viscosity and thermal diffusivity, respectively, Φ is the dissipation function, and qw ¼ k½∂T=∂x3 �w is the surface heat flux which is positive when heat enters into fluid from the surface. The term ð∂ =∂t ar2 Þqw in Eq. (9) is interpreted as a source term in the formal diffusion process of heat flux on the surface. To extract a skin friction field from a surface temperature field, fQ should be modeled. In TSP measurement on a pre-heated or pre-cooled body, a TSP layer is usually coated on a thin white base layer on a body, resulting a composite polymer later (a TSP layer plus a base layer). This arrangement enhances not only surface thermal signatures but also the luminescent emission from the TSP. A heuristic model is fQ ¼ γ ðTref Tw Þ, where γ is an empirical heat transfer coefficient, Tw is the TSP surface temperature, and Tref is the reference temperature inside the body (a suitable refer­ ence temperature selected for a specific experiment). This model was used in global skin friction diagnostics using TSP [50,51]. Relation between skin friction and surface scalar concentration can be derived from the binary mass diffusion equation with a source term, i. e., ∂φ1 =∂t þ u⋅rφ1 ¼ D12 r2 φ1 þ Qs [52,53], where φ1 ¼ ρ1 =ρ is the relative concentration (density) of the species 1, ρ ¼ ρ1 þ ρ2 is the total density of the binary gas (a mixture of species 1 and 2), D12 is the diffusivity of a binary system, and Qs is the source term. Since this diffusion equation has the same mathematical structure as the energy equation, we have an on-wall relation on a flat surface

τ ⋅ rφ1 ¼ μfM ; where the virtual source term on a surface is written as � � � � � 3 � 1 ∂ ∂Qs ∂ φ1 þ D12 ; fM ¼ D12 r2 m_ 1w þ D12 ρ ∂t ∂x3 w ∂ðx3 Þ3 w

Fig. 9. Skin-friction lines mapped on the surface of the wing-body junction model viewed from (a) the suction side and (b) pressure side of the wing for AoA ¼ 6� , where the color map shows the luminescent intensity distribution of the oil film. From Zhong et al. [44]. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

(10)

(11)

and m_ 1w ¼ D12 ρ½∂φ1 =∂x3 �w is the surface diffusive flux of species 1. The term ð∂ =∂t D12 r2 Þm_ 1w in Eq. (11) is interpreted as a source term in the formal diffusion process of mass flux on the surface. How to model fM depends on a specific measurement technique for species. For mass transfer visualization with PSP, a model for fM is given by fM ¼ ρw γm ðφ1w φ1;00 Þ, where φ1w and φ1;00 are the values of φ1 at the gasPSP interface and the PSP-solid interface, respectively, γm ¼ 2D1p =h is a coefficient, D1p is the diffusivity of the species 1 in the polymer, and h is the PSP coating thickness. For a curved surface, Eqs. (8) and (10) can be derived by using the method of differential geometry, where additional terms related to the surface curvature are included [59].

3. Skin friction from surface heat and mass transfer visualization 3.1. Skin friction, surface temperature and scalar concentration Relation between skin friction and heat transfer is generally known as the Reynolds analogy. The functional form of the Reynolds analogy between local skin friction coefficient and local Nusselt number can be obtained for laminar boundary layers from a class of similar solutions. Solving the energy equation for a boundary layer, Lighthill [57] derived an analytical relation between local skin friction and surface heat flux from a small heated element. This explicit relation provides a theoretical foundation for a flush-mounted hot-film skin friction sensor [58]. However, local solutions for an isolated heated element are not appli­ cable to globally heated or cooled surfaces in complex flows such that direct conversion from measured local heat flux to skin friction is not possible. Global heat-transfer-based skin friction measurement methods are desirable, which are particularly relevant to global surface temper­ ature measurements with temperature sensitive paint (TSP) [50].

3.2. Variational formulation For generality, Eqs. (8), (10) and (16) are written in a generic form G þ τ ⋅rg ¼ 0;

(12)

where g and G are the generic measurable quantities, which are defined differently depending on specific visualization technique used in ex­ periments. For example, G ¼ μfQ and g ¼ Tw for heat transfer visuali­ zation, and G ¼ μfM and g ¼ φ1 for binary mass transfer visualization. By minimizing the following functional with a smoothness regulariza­ tion term in an image domain D 8

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Fig. 10. Zoomed-in views of skin-friction lines: (a) front region on the wingtip surface (Region A in Fig. 8), (b) a long separation bubble on the suction surface (Region B in Fig. 8), and (c) a short separation bubble on the suction surface (Region D in Fig. 8). From Zhong et al. [44].

Z ðG þ τ ⋅rgÞ2 dx1 dx2 þ α

Jðτ Þ ¼ D

Z � � 2 2 jrτ1 j þ jrτ2 j dx1 dx2 ; D

the Neumann condition ∂τ =∂n ¼ 0 imposed on a domain boundary ∂D. Since Eqs. (8) and (10) are valid instantaneously, unsteady skin friction fields can be extracted from unsteady surface temperature and scalar measurements. In an error analysis, substitution of the decompositions g ¼ g0 þ δ g, G ¼ G0 þ δ G and τ ¼ τ 0 þ δτ to Eq. (14) yields an error propagation equation, where δ g, δ G and δτ are errors, and g0 , G0 and τ 0 are the nonperturbed fields that exactly satisfy Eq. (14). A formal estimate of the relative skin friction error ðδτ ÞN ¼ δτ ⋅N T is [50].

(13)

the Euler-Lagrange equations are obtained, i.e., ½ G þ τ ⋅ rg � rg

α r2 τ ¼ 0;

(14)

where α is a Lagrange multiplier, and r ¼ ∂=∂ xi and r2 ¼ ∂2 = ∂ xi ∂ xi (i ¼ 1; 2) are the gradient operator and Laplace operator, respectively. Given G and g, Eq. (14) can be solved numerically for τ ¼ ðτ1 ; τ2 Þ with 9

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δG krg0 k kτ 0 k

� � τ0 α 2 ðδτ ÞN r ⋅δNT þ ; kτ 0 k kτ 0 k krg0 k2

�

�

(15)

� � where kτ 0 k is a mean value of skin friction, and NT ¼ rg0 =�rg0 � is the unit normal vector to an iso-value line g0 ¼ const:. The first term in the RHS of Eq. (15) is the contribution from the elemental error in mea­ surement of G. The second term is the contribution from the elemental error in measurement of the surface gradient of the relative intensity. The third term is the contribution from the artificial diffusion of ðδτ ÞN associated with the Lagrange multiplier. Since the first term in the RHS � � 1 of Eq. (15) is proportional to �rg0 � , the relative error ðδτ ÞN = kτ 0 k � � increases as �rg0 � decreases. The third term is proportional to � � 2 α�rg0 � , indicating that the Lagrange multiplier α must be sufficiently � � small to reduce the error particularly when �rg0 � is small. 3.3. Example 3: Circular cylinder in water flow TSP measurements on a circular cylinder were conducted in the cavitation tunnel at CEIMM (CNR-INSEAN at Rome, Italy), which is a closed-loop water tunnel having a 1:5.96 contraction nozzle and a square test section of side of 600 mm [51]. Fig. 11 shows the experi­ mental set-up at CEIMM cavitation tunnel, where UV LED lamps lighted up the TSP-coated cylinder in cross-flow and a fast-camera recorded images from the cylinder’s surface at the TSP emission wavelength (615 nm). Free-stream turbulence intensity and flow uniformity at the channel centerline were 1.5% and 0.4%, respectively. A hollow aluminum cylinder model was used in experiments, and the cylinder diameter, length and thickness were 36 mm, 600 mm and 13.5 mm, respectively. The experiments were conducted at free stream velocity ranging from 2 m/s to 4 m/s. The Reynolds numbers, based on cylinder diameter and water kinematic viscosity at 25 � C ranged from 72,000 to 144,000. The cylinder was uniformly heated up by running hot water through its hollow core. TSP was coated on a white base coating on the cylinder surface, and thus surface temperature signatures generated by flow structures were clearly visualized. A time sequence of TSP images were obtained by using a fast CCD camera, and temperature data on the cylinder surface were quantitatively extracted from TSP images using a TSP calibration relation. Fig. 12 shows a surface temperature field mapped on the surface mesh of the cylinder. Fig. 13(a) shows a time-averaged surface temperature image of the mid-section of the cylinder. From this surface temperature field, a skin

Fig. 12. Surface temperature field mapped on the surface mesh of the cylinder, viewing position of the camera, and the coordinate system. From Miozzi et al. [51].

friction field is extracted by solving Eq. (14) where G ¼ μfQ and g ¼ Tw . For a heated cylinder in water flow, a model fQ ¼ γ ðTref Tw Þ is used, where Tref is the body temperature of the heated cylinder, and γ is an empirical heat transfer coefficient (treated as a constant in this case). Fig. 13(b) shows the extracted skin friction topology, indicating that the primary laminar flow separation occurs at the angular position φ � 80o and re-attachment due to transition occurs at φ � 95o such that a sep­ aration bubble forms. The secondary separation occurs immediately after the re-attachment at φ � 100o . The re-attachment and secondary separation lines are so close that they could not be clearly distinguished in surface temperature images without detailed image processing. In this quasi-2D flow, the separation and attachment lines can be modeled as lines composed of many alternatively distributed nodes and saddles (referred to as singular lines). If a rectangular boundary is selected to enclose a segment of a separation line or an attachment line, the P-B formula gives a topological constraint #N #S ¼ 1 since there is no switch point on the boundary. Quantitative results of skin friction distribution along the cylinder’s circumference were obtained by taking the spatial average of skin fric­ tion in the spanwise direction. Fig. 14 shows comparison with mea­ surement data at Red ¼ 6000 [60] and Red ¼ 100,000 [61], and with LES data at Red ¼ 10,000 [62]. Schlichting’s series solution for laminar boundary layer [63] is also plotted in Fig. 14 for further reference. All data are normalized by their maximum. The analytical solution deviates from the other data just after the maximum of skin friction has been reached, while all the other profiles collapse quite well across a large part of the laminar region. Close to the laminar separation line (jτ j ¼ 0), only the data from Ref. [61] agrees with the TSP-derived results up to φ � 80o . Downstream of this position (at larger φ), the TSP-derived relative skin friction profile captures the re-attachment and secondary separation lines (here defined as the zero crossings of jτ j). Further, it is found that the re-attachment and secondary separation lines drift downstream and upstream at the vortex shedding frequency. The high spatial resolution data allow a deeper analysis of the evolution of the re-attachment and secondary separation, revealing critical features in phase averaged skin friction fields [51].

Fig. 11. Experimental set-up at CEIMM cavitation tunnel, where UV LED lamps light up the TSP-coated cylinder in cross-flow and a fast-camera records pic­ tures from the cylinder’s surface at the TSP emission wavelength (615 nm). From Miozzi et al. [51]. 10

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Fig. 13. (a) Time-averaged surface temperature image, and (b) extracted time-averaged skin friction lines. Text arrows identify the laminar separation line and the secondary turbulent reattachment/separation sequence. Markers exactly replicate the same positions in both maps to facilitate comparison. From Miozzi et al. [51].

reference conditions, respectively, A ¼ ð1 þ Kρ2ref Þ 1 and B ¼ Kρ2ref A are the Stern-Volmer coefficients for aerodynamics applications, and ρ2ref is the value of ρ2 at the wind-off reference condition. When nitrogen is added in flow for surface mass-transfer visualization with PSP, the relative density of nitrogen (the species 1) in PSP is ρ1 ¼ 1 ρ2 . In this case, in Eq. (12), g ¼ Iref =I is the normalized image intensity, τ is the relative skin friction vector projected onto the image plane, and G � g Iref =I00 is the virtual source term, where I00 is the luminescent intensity of PSP that is corresponding to ρ1;00 at the PSP-solid interface. For PSP in flow added with nitrogen, G < 0 since the concentration of nitrogen at the PSP-solid interface is smaller such that I > I00 ,. Fig. 16 shows normalized PSP intensity ratio images (I=Iref ) in the dual colliding impinging jets with the offsets of 0, 4, 7.5, and 11.5 mm, respectively. Skin friction fields are extracted from these images by solving Eq. (14). There are 123 � 186 vectors in each skin friction field. Skin friction lines are shown in Fig. 17. As shown in Fig. 17(a), the impingement regions of the two head-on jets are merged such that there is only one node near the middle and the merged jet is bifurcated in the vertical direction (y-axis). When the two jets are offset by 4 mm, as visualized in Fig. 17(b), two wall-jets are deflected at the inclination angle of about 25o relative to the y-axis. Two nodes associated with the impinging jets are clearly separated and a saddle occurs between them. As the offset is increased to 7.5 and 11.5 mm, the separation between the two nodes is enlarged while the saddle remains at the center between them [see Fig. 17(c)–(d)].

Fig. 14. Comparison between present and previous time averaged relative skinfriction distributions, continuous line: present data), circle: Rizzetta and Visbal [62], square: Olson et al. [60], Δtriangle: Achenbach [61], *: Schlichting et al. [63]. From Miozzi et al. [51].

3.4. Example 4: Dual colliding impinging nitrogen jets To examine the capability of mass-transfer-based skin friction di­ agnostics, PSP visualizations in dual colliding impinging nitrogen jets were conducted [52]. Fig. 15 shows experimental setup for dual colliding impinging nitrogen jets. Two nitrogen jets impinged on a wall toward each other from two straight tubes with an inner diameter of D ¼ 5:18 mm at an impingement angle of φ ¼ 30o . The nozzle-to-surface distance was H ¼ 29:5 mm and the separation between the exit centers of the two tubes along the x-axis was 80 mm. The offset between the two tubes in the y-axis was adjusted to generate different flow patterns. The nitrogen jet exit velocity was 4.14 m/s. For surface mass-transfer visualization with PSP, Eq. (10) was rewritten for the measured luminescent intensity. The luminescent emis­ sion of PSP is related to the relative oxygen density ρ2 (the species 2) by the Stern-Volmer relation. The Stern-Volmer relation is expressed as Iref = I ¼ A þ Bðρ2 =ρ2ref Þ or ρ2 ¼ ðρ2ref =BÞðIref =IÞ Aρ2ref =B, where I and Iref are the luminescent intensities of PSP in the wind-on and wind-off

3.5. Example 5: Shock/boundary-layer interaction Sublimating chemicals like naphthalene and acenaphthene have been used for mass transfer measurements [64,65]. Sublimation mass-transfer visualizations were conducted by using a mixture of ace­ naphthene with diethylether on the flat plate in shock/boundary-layer interaction over a vertically-mounted triangular fin with the leading-edge swept angle of 70� at the angle of attack of 40� and Mach 6 in Von Karman Institute of Fluid Mechanics [64,65]. A sublimation coating was applied on a black-painted surface for mass transfer visu­ alization. A typical sublimation visualization image is shown in Fig. 18 (a). The thickness of a sublimating coating is reduced during a run, visualizing local mass transfer. For a thin sublimation layer on a wall, the time rate of the layer thickness h is described by dðρ10 hÞ=dt ¼ m_ 1w;s Hðt t0 Þ, where ρ10 is the density of a sublimating material in the 11

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Fig. 15. Experimental setup for dual colliding impinging jets, (a) side view and (b) top view. From Liu et al. [52].

solid state (or the saturated vapor state), and Hðt t0 Þ is the Heaviside function. Integration of this equation leads to the mass flux of the sub­ limating material, m_ 1w;s ¼ ρ10 ðΔh =ΔtÞ, where ΔT ¼ t1 t0 is a time span, Δh ¼ href h is a thickness change, and h and href are the layer thicknesses at t ¼ t1 and t ¼ t0 , respectively. On the other hand, the mass transfer coefficient is defined as c1m ¼ m_ 1w;s =ρw ρ1w;s , and therefore

field under certain conditions such that near-wall flow structures could be reconstructed. These questions are relevant to global skin friction diagnostics based on surface pressure measurements with PSP. An attempt to investigate these problems has been made [55]. Relation between skin friction vector and surface pressure can be derived from the Navier-Stokes equations in a general surface coordi­ nate system [55,59]. This on-wall relation is written as

ρ1w;s ¼ c1m1 ðΔh =ΔtÞ. In image-based measurements, the sublimating coating thickness is proportional to the light radiance scattered from the coating, i.e., h=href ¼ I=Iref . Therefore, in Eq. (12), g ¼ I= Iref is the normalized image intensity, τ is the projected skin friction vector, and G � g þ 1 is the virtual source term. A skin friction field with 209 � 358 vectors is extracted by solving Eq. (14) from the single image in Fig. 18(a). Extracted skin friction lines are shown in Fig. 18(b). On the pressure side of the fin where strong shock/boundary-layer interaction occurs on the floor, the primary and secondary separation lines (denoted by S1 and S2 ) and the attachment line (denoted by A1 ) are clearly identified. These topological features are similar to those given by CFD predictions and surface oil visualizations at Mach 4 and the angle of attack of 30.6� [66]. On the lee side of the fin where the floor boundary layer interacts with the expansion waves and separated flow from the leading edge of the fin, skin friction lines reveal multiple separation and attachment lines.

τ ⋅ rp ¼ μfΩ ; where a virtual source term fΩ is expressed as � � ∂Ω fΩ ¼ μ μωw ⋅ K ⋅ ωw þ μθ ðωw � nÞ⋅rθw ; ∂n w

(16)

(17)

where Ω ¼ jωj2 =2 is the enstrophy, ∂=∂n is the derivative along the wallnormal direction, ω ¼ r � u is the vorticity, K is the surface curvature tensor, θ ¼ r⋅u is the dilation rate, μ is the dynamic viscosity, μθ is the longitudinal viscosity, and n is the unit normal vector of the surface. The subscript w in the variables and operators in Eq. (17) denotes the quantities on a wall. Eq. (16) represents a formal balance between rp projected on a skin friction vector τ and the scalar quantity fΩ that is originated from the diffusion term in the Navier-Stokes equations. In Eq. (17), the first term μ½∂Ω=∂n�∂B is the boundary enstrophy flux (BEF), and the second term is interpreted as a curvature-induced contribution. The term ωw ⋅K⋅ωw in Eq. (17) is formally interpreted as the interaction between the surface curvature and the vorticity on the surface. By considering the geometry of a vorticity line on the surface (referred to as a boundary vorticity line that is a tangent line to the vorticity ωw on the surface), another expression is fΩ ¼ μ½∂Ω=∂n�∂B 2μκω Ω∂B ðn ⋅nω Þ, where κω is the curva­ ture of the boundary vorticity line, and nω is the principal unit normal vector of the boundary vorticity line on the surface [55]. The third term is interpreted as a contribution induced by the temporal-spatial change

4. Skin friction from surface pressure visualization 4.1. Skin friction and surface pressure An important implication of the Taylor-series solution of the NS equations, Eq. (2), is that near-wall flow structures are solely determined by skin friction and surface pressure. A question is whether skin friction is intrinsically related to surface pressure. A further question is whether a skin friction field could be extracted from a measured surface pressure 12

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Fig. 16. PSP intensity ratio images of the dual colliding impinging jets with the offsets of (a) 0, (b) 4, (c) 7.5, and (d) 11.5 mm. From Liu et al. [52].

of the fluid density on the surface. When the Reynolds number is suffi­ ciently large, the second term related to the surface curvature could be neglected. At a wall, the third term related to the compressibility could be neglected. Therefore, for a flat surface, fΩ is just the BEF.

initially neglected. In this case, the perturbed skin friction field denoted

by τ ð1Þ can be described by the approximate equation τ ð1Þ ⋅rpð1Þ ¼ μf Ω . Therefore, a τ ð1Þ -field can be obtained by solving the Euler-Lagrange ð0Þ

equations, Eq. (14), where G ¼ μf Ω is known for the base flow. Further, a perturbed near-wall velocity field can be reconstructed by using Eq. (2) based on the τ ð1Þ - and pð1Þ -fields. This approximate method has been used to reconstruct typical separation flow structures [27]. A scheme of implementing the above approximate method is described as follows. ð0Þ

4.2. Variational formulation Since Eq. (16) has the same form as Eqs. (8) and (10), the variational formulation described in Section 3.2 is applicable, and thus the EulerLagrange equations, Eq. (14), can be solved to extract skin friction fields from surface pressure fields, where G ¼ μfΩ and g ¼ p. In principle, to solve this inverse problem, a fΩ -field must be given. How­ ever, for complex flows, the BEF is not generally known since it is a more difficult quantity with the second-order velocity derivatives to measure or compute accurately. Therefore, a fΩ -field should be suitably modeled or approximated. An approximate method was proposed to obtain a skin friction field induced by a surface-pressure variation imposed on a base flow [27]. In a zeroth-order approximation, a known base flow is considered, which

(1) The base flow is selected for a specific application, where pð0Þ and

f Ω are known. (2) The surface-pressure variation δp is considered. In simulation, a δp-field can be reconstructed by linearly superposing some elemental δp-patterns. In experiment, a δp-field can be obtained from global pressure measurements. (3) The perturbed surface-pressure field is given bypð1Þ ¼ pð0Þ þ δp. (4) The τ ð1Þ -field is obtained by solving Eq. (14) with the given pð1Þ ð0Þ

satisfies τ ð0Þ ⋅rpð0Þ ¼ μf Ω , where the superscript ‘0’ denotes the base flow. A base flow could be a relatively known steady attached flow (such as boundary layer). Then, we consider a physical situation where a surface pressure variation is suddenly imposed on the base flow through certain physical process (such as impinging shock wave and moving vortex). A perturbed surface pressure field on a surface is expressed by a decomposition pð1Þ ¼ pð0Þ þ δp, where δp is a surface-pressure variation that is a function of the location and time. A specific form of δp is written as δp ¼ gðxÞHðtÞ, where gðxÞ defines a spatial distribution and HðtÞ is a Heaviside function.

and G ¼ Rel 1 f Ω . (5) The near-wall velocity field is reconstructed by using Eq. (2) based on the extracted τ ð1Þ - and pð1Þ -fields.

ð0Þ

ð0Þ

4.3. Example 6: Square-plate junction The feasibility of extracting a skin friction field from a surface pressure field was validated when a BEF field is given in a junction flow over a square cylinder [55]. PSP measurements were conducted in the Tohoku-University Basic Aerodynamic Research Wind Tunnel (T-BART) [67]. This is a suction-type wind tunnel that has a solid wall test section of 300 mm width, 300 mm height and 760 mm length. In junction flow

As an approximation, a f Ω -field in the base flow is not affected by δp ð0Þ

in a short time, and in other words the interaction between f Ω and δp is ð0Þ

13

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Fig. 17. Skin friction lines of the dual colliding impinging jets with the offsets of (a) 0, (b) 4, (c) 7.5, and (d) 11.5 mm. From Liu et al. [52].

measurements, the test model was a 3D square cylinder that has 40 � 40 mm cross-section and 100 mm height, as shown in Fig. 19. The test model was vertically mounted on the flat plate and could be rotated by a turntable. PSP measurements were conducted mainly on the floor around the model. The free-stream velocity was set at 50 m/s in PSP measurement. The incident angle relative to the free-stream was set at 0� for the square cylinder. The Reynolds number based on the model length was ReD ¼ 1.3 � 105 for the square cylinder and 1.8 � 105 for the diamond cylinder. The local Reynolds number is Rex ¼ 7.8 � 105 for the location of the front of the cylinder at 230 mm from the flat-plate leading edge. It was confirmed by hot-wire measurement that the incoming boundary layer was laminar state under these conditions. Fig. 20(a) shows a normalized surface pressure field obtained from PSP measure­ ments in this junction flow. For comparison, GLOF skin friction diagnostics were conducted at the same test conditions, where perylene-mixed silicone oil was used [67]. Fig. 20(b) shows a typical GLOF image in the junction flow. Fig. 21 (a) shows skin friction lines extracted from GLOF images. From this extracted skin friction field and the surface pressure field obtained by

using PSP, a BEF field (a fΩ -field) is reconstructed by using the relation fΩ ¼ μ 1 τ ⋅rp. Fig. 21(b) shows the BEF field reconstructed from PSP and GLOF measurements. Some subtle features are marked in Fig. 21(b), particularly nulls corresponding to isolated critical points. Then, from the surface pressure field and BEF field, a skin friction field is extracted by solving Eq. (14). Fig. 22 shows extracted skin friction vectors and lines, which are in good agreement with the results extracted from GLOF images. The relative error in the whole measurement domain is less than 7%. Fig. 22(b) shows interesting skin friction topology on the floor sur­ face. A saddle S1 is located at the upstream of the square cylinder, from which the primary necklaced separation line is originated. In addition, attachment lines are originated from the sides of the cylinder. The pri­ mary separation and attachment lines are associated with a single large horseshoe vortex forming in the front of the cylinder. Behind the cyl­ inder, a combination of the saddle (S2 ) and the spiraling sink nodes (foci) N1 and N2 are observed, which are the time-averaged on-wall footprints of the shedding wake structures. The extracted topology in this junction flow is consistent with the P-B formula [13]. 14

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method described in Section 4.2, the Falkner-Skan flow is considered as

a base flow, where pð0Þ and f Ω are known. To reconstruct elemental ðτ ; pÞ-structures, several simple forms of δp are considered. An elemental form is a Gaussian function, i.e., δp ¼ g0 expð r2 =2σ 2 Þ, where r ¼ jr rc j is the distance from the pressure center r ¼ ðxc ; yc Þ, g0 is the ð0Þ

Fig. 18. (a) Sublimation image (from Zemsch [64], Zemsch et al. [65]), and (b) extracted skin friction lines on the flat plate in shock/boundary-layer interac­ tion over a vertically-mounted triangular fin with the angle of attack of 40� at Mach 6 (flow is from left to right). From Liu et al. [52].

4.4. Example 7: Elemental separation structures For a base flow with a given BEF field (a fΩ -field), skin friction and surface pressure are coupled according to Eq. (16), which determine near-wall flow structures. Such a coupled structure is called ðτ ; pÞ-structure [27]. To demonstrate the application of the approximate

Fig. 20. PSP and GLOF visualizations in the junction flow over a square cyl­ inder: (a) normalized surface pressure filed obtained from PSP, and (b) GLOF image. From Liu et al. [55].

Fig. 19. Schematic of the junction flow over a cylinder on a flat plate. From Kakuta et al. [67]. 15

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Fig. 21. Skin friction and BEF fields in the junction flow over a square cylinder: (a) skin friction lines extracted from GLOF images, and (b) BEF field. From Liu et al. [55].

strength, and the standard deviation σ defines the influential region. When the base flow is perturbed by a local high-pressure variation with g0 > 0, skin friction lines exhibit a source in a reconstructed skin friction field (a τ ð1Þ -field). In contrast, for g0 < 0, a local low-pressure variation leads to a sink in a τ ð1Þ -field. Another elemental form is a helical surface-pressure variation given by δp ¼ γ0 arc tanðθÞ, where θ is the polar angle and γ0 is the strength. This distribution of δp generates a counter-clockwise vortex in a τ ð1Þ -field. Complex near-wall flow structures can be reconstructed by super­ position of some elemental forms of δp. For example, a high-pressure Gaussian variation and two helical surface-pressure variations are su­ perposed, and the resulting δp-field is applied to the base flow to form a pð1Þ -field, as shown in Fig. 23(a). The reconstructed τ ð1Þ -field by solving Eq. (14) based on the pð1Þ -field is shown in Fig. 23(b). The source node is marked as N1 . The vortex pair centers (foci) are marked as N2 and N3 , and the saddle S2 is induced. The saddles S1 and S3 appear upstream and downstream, respectively. The front segment of the envelope is a sepa­ ration line originated from S1 , and the back segment of the envelope is an attachment line connecting to S3 . The distribution of these critical points in Fig. 23(b) satisfies the topological constraint #N #S ¼ 0 that

Fig. 22. Skin friction field extracted from the surface pressure field and BEF field in the junction flow over a square cylinder: (a) skin friction vectors, and (b) skin friction lines. From Liu et al. [55].

is a special form of the P-B formula for this flow, where #N and #S denote the numbers of nodes and saddles, respectively. This ðτ ; pÞ-structure corresponds to the observed separated flow over a generic hill model [41], and particularly the vortex pair pattern is known as the owl-face of the first kind [20]. Then, the corresponding near-wall ve­ locity field is reconstructed by using Eq. (2). Fig. 23(c) and (d) show near-wall streamlines and contours of the enstrophy near the wall, respectively. Streamlines near the vortex centers spiral upward due to the presence of low-pressure regions there. This example demonstrates a convenient semi-analytical approach for reconstructing complex near-wall flows by superposing some specific δp-fields. It is advantageous that a linear superposition could be applied approximately to the formation of δp-patterns since the pressure perturbation is reasonably described by a linearized wave equation. It is emphasized that the elemental τ -structures (source, sink and vortex) 16

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Fig. 23. Hill model structure in a skin friction field, (a) the normalized pð1Þ -field with iso-pressure lines, (b) skin friction lines overlaid on the pð1Þ -field, (c) near-wall streamlines, and (d) contours of the enstrophy near a wall. The freestream flow is aligned with the y-direction in the image plane. From Liu [27].

cannot be simply and directly superposed in a linear fashion. Appar­ ently, the τ -structures in Fig. 23(b) are similar to the potential-flow patterns obtained by linearly superposing the elemental solutions of the Laplace equation. However, the elemental τ -structures are funda­ mentally different from the potential-flow counterparts, which are reconstructed by using the variational method based on the nonlinear coupling relation between the two dynamical quantities ðτ ; pÞ in a viscous flow. In this sense, these structures represent the approximate near-wall solutions of the NS equations because both Eq. (2) and Eq. (16) are directly derived from the NS equations.

film equation by using the projection method. This relation has the same mathematical form as the optical flow equation for global velocity di­ agnostics. For surface heat and mass transfer visualizations, the relations between skin friction and surface temperature and scalar concentration are derived from the energy equation and mass transport equation, respectively, and they can be expressed in a reduced form of the optical flow equation. Further, the relation between skin friction and surface pressure is derived from the Navier-Stokes equations, and interestingly it has the same form as those for surface heat and mass transfer visualizations. In principle, to extract skin friction fields from surface visualization images as an inverse problem, a variational method is applied to these relations, and the Euler-Lagrange equations are obtained. The standard finite difference method is used to solve the Euler-Lagrange equations to determine skin friction fields in the image plane. From a successive pair of GLOF images, a snapshot solution is obtained, which captures local skin friction signatures at that moment in an oil-film evolution. Then, a time-averaged skin friction field is reconstructed by averaging sequen­ tial snapshot solutions. Therefore, the GLOF method gives a timeaveraged (or steady) skin friction field since the oil-film evolution is a slow process. In contrast, the methods based on surface heat and mass transfer and pressure visualizations can be used for unsteady global skin

5. Conclusions High-resolution skin friction fields in complex flows can be extracted from conventional surface flow visualizations with luminescent oil, temperature sensitive paint (TSP), pressure sensitive paint (PSP) and sublimation coating. The foundations of such global skin friction di­ agnostics are the coupling relations between skin friction and measured quantities (oil-film thickness, surface temperature, surface scalar den­ sity, surface pressure, and sublimation coating thickness). For global luminescent oil-film (GLOF) visualization, the relation between skin friction and luminescent intensity of an oil is derived from the thin-oil17

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friction diagnostics when the virtual source terms in the corresponding relations are suitably modeled or approximated. From a perspective of image processing, these methods for different surface flow visualizations are incorporated into a unified framework of the variational methods. These image-based measurement techniques are mature and ready to implement in various facilities. The methods described in this paper have been used for global skin friction di­ agnostics in complex flows, and topological features are examined, including isolated critical points and separation and attachment lines. The Poincare-Bendixson index formula is a useful tool to evaluate the topological consistency of extracted skin friction fields.

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Iref : luminescent intensity at reference condition (radiance) K: surface curvature tensor m_ 1w : mass diffusive flux of species 1 at wall (kg-s 1-m 2) n: unit normal vector of surface p: pressure (N-m 2) qw : heat flux at wall (W-m 2) t: time (s) us : non-dimensional velocity parallel to surface (m-s 1) u3 : non-dimensional velocity normal to surface (m-s 1) x1 ; x2 : image coordinates (m or pixel) xi : normalized coordinates (i ¼ 1; 2; 3) X1 ; X2 : surface coordinates in object space (m)Greeks α: Lagrange multiplier δp: surface pressure variation (Pa) λ: scaling factor in projection μ: viscosity (Pa-s) ρ: gas density (kg-m 3), ρ1 þ ρ2 ρ1 : density of species 1 (kg-m 3) ρ2 : density of species 2 (kg-m 3) τ: skin friction vector (N-m 2) bτ : projected and scaled skin friction vector φ1 : relative density of species 1 φ2 : relative density of species 2 ω: vorticity (s 1) Ω: enstrophy (s 2)Subscript o: oil ref: reference condition w: wall condition

Nomenclature D12 : diffusivity of a binary gas (m2-s 1) fI : source term defined in Eq. (4) fQ : source term defined in Eq. (9) fM : source term defined in Eq. (11) fΩ : source term defined in Eq. (17) g: normalized measurable quantity gI : normalized luminescent intensity gI ¼ I=Iex G: source term in Eq. (12) h: coating thickness (m) HðtÞ: Heaviside function I: luminescent or light intensity (radiance) Iex: excitation intensity (radiance) I0 : luminescent intensity at zero-oxygen condition (radiance) I00 : luminescent intensity of PSP at the PSP-solid interface

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