The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Backward-Facing Step (BFS) flow is one representative model for separation flows, which can be widely seen in aerodynamic flows (airfoil, spoiler, high attack angle process), engine flows, condensers, vehicles (cars, boat), heat transfer systems, and even the flow around buildings, etc. The flow separation after a simple stage will introduce separation bubble formation, evolution and re-attachment process, which is dependent on the BFS geometric design, the inlet and outlet conditions, turbulent intensity, as well as heat transfer conditions. In the past decades, it has been widely studied by various theoretical, experimental and numerical methods. Considering the importance of BFS flow in both theoretical and engineering aspects, this paper is focused on a review study of BFS flows from fundamental understandings to various experimental and numerical developments in a historical viewpoint. Basic models and the parameter-based after-step flow laws are summarized and categorized in this study. It is shown that the step size (duct expansion ratio) will define the basic re-circulation and re-attachment process, while the coupled effects of inflow parameters and the perturbation designs also help shape the flow behaviors after BFS. The review is also extended with model generalizations and the implications on system design, especially the heat transfer effects and the representative control designs are discussed in detail. Future trends and prospects in BFS studies are also included in this study.

A review of Backward-Facing Step (BFS) flow mechanisms, heat transfer and control

Bio-inspired, intelligent flexible sensing skin for multifunctional flying perception

A comprehensive review of all relevant experimental data was completed, including recent data for the drag coefficient for a sphere in supersonic and hypersonic flows. The primary characterization ...

https://arc.aiaa.org/doi/pdf/10.2514/1.J060153

Supersonic and Hypersonic Drag Coefficients for a Sphere

The synthesis of novel Eu complexes with 1,3-di(thienyl)propane-1,3-diones as ligands as well as their luminescence properties in different polymers are reported. The new temperature sensitive paints (TSPs) exhibit not only exceptional high temperature sensitivity over a wide range of temperatures but are also characterized by negligible pressure sensitivity and marked photostability. This is why they are outstandingly suitable for applications, e.g. in aerodynamics and hydrodynamics.

Europium 1,3-di(thienyl)propane-1,3-diones with outstanding properties for temperature sensing

We study the Laminar Separation Bubble (LSB) which develops on the suction side of a NACA 0015 hydrofoil by means of a Temperature-Sensitive Paint (TSP), at a Reynolds number of $$1.8\times 10^5$$
 and angles of attack $$\mathrm{AoA} = [3^{\circ }$$
 , $$5^{\circ }$$
 , $$7^{\circ }$$
 , $$10^{\circ }$$
 ]. The thermal footprints $$T_\mathrm{w}(x,y,t)$$
 of the fluid unveil three different flow regimes whose complexity in time and space decreases when $$\mathrm{AoA}$$
 increases, up to $$10^{\circ }$$
 where the LSB-induced spatial gradients are linked to quasi-steady positions in time. At $$\mathrm{AoA} =7^{\circ }$$
 the LSB system undergoes a 3D destabilization, that induces C-shaped arcs at separation and weak bubble-flapping at reattachment. Structural changes occur at $$AoA=5^{\circ }$$
 and $$3^{\circ }$$
 : bubble-flapping raises homogeneously at reattachment while intermittent, wedge-shaped events alter the LSB shape. The relative skin-friction vector fields $$\varvec {\tau }_\mathrm{w}(x,y,t)$$
 , extracted from $$T_\mathrm{w}(x,y,t)$$
 by means of an optical-flow-based algorithm, provide the topology of the flow at the wall and feed a physics-based criterion for the identification of flow separation $${\mathfrak {S}}(y,t)$$
 and reattachment $${\mathfrak {R}}(y,t)$$
 . This criterion fulfills, in average, a novel skin-friction ground-truth estimation grounded on the determination of the propagation velocity of temperature fluctuations. The obtained $${\mathfrak {S}}(y,t)$$
 is composed of several manifolds that extend spanwise from saddle points to converging nodes via the saddles unstable manifold, while, at least at higher AoA, manifolds that compose $${\mathfrak {R}}(y,t)$$
 move from diverging nodes to saddle points via the saddles stable manifolds. The triggering of a wedge-shaped event by a rising $$\varOmega$$
 -shaped vortex in the reverse LSB flow is captured and described in analogy to a simplified model.

Skin friction and coherent structures within a laminar separation bubble

Tests were conducted at the Cryogenic Ludwieg-Tube Gottingen to investigate the influence on onset of boundary-layer transition of thermal gradients on the model surface, whereby the temperature-sensitive paint technique has been used to detect transition. The laminar airfoil under investigation was tested at high Reynolds and at both low and high subsonic Mach numbers. The influence on boundary-layer transition of the temperature difference between the flow and the model surface due to the temperature drop occurring during a test run at the Cryogenic Ludwieg-Tube Gottingen was examined. Various combinations of flow conditions and angles of attack were employed to investigate different stability situations. Linear-stability computations, based on the experimental data, were also carried out. The unfavorable impact of surface temperatures larger than the adiabatic-wall temperature was identified to be dependent on the considered boundary-layer-stability situation. For certain stability situations, it was s...

Nonadiabatic Surface Effects on Transition Measurements Using Temperature-Sensitive Paints

Experimental Investigation of the Effect of Forward-facing Steps on Boundary Layer Transition

The influence of the streamwise pressure gradient and a nonadiabatic surface on boundary layer transition was experimentally investigated at the Cryogenic Ludwieg-Tube Gottingen, Germany. Boundary layer transition was detected nonintrusively by means of the temperature-sensitive paint technique. The wind-tunnel model was designed to achieve a quasi-uniform streamwise pressure gradient over a large portion of the model chord length. This allowed the effects on boundary layer transition of the streamwise pressure gradient and wall temperature ratio to be decoupled. The model was tested at high Reynolds numbers and at a high subsonic Mach number. Favorable, almost-zero, and adverse streamwise pressure gradients were considered; and various temperature differences between the flow and the model surface were implemented. Stronger flow acceleration and lower wall temperature ratios led to an increase of the transition Reynolds number. Larger increases in the transition Reynolds number were obtained at more pron...

Pressure Gradient and Nonadiabatic Surface Effects on Boundary Layer Transition

The influence of unit Reynolds number (
 $$Re_1=17.5\times 10^{6}$$
 –
 $$80\times 10^{6}\,\text {m}^{-1}$$
 ), Mach number (
 $$M= 0.35$$
 –0.77) and incompressible shape factor (
 $$H_{12} = 2.50$$
 –2.66) on laminar–turbulent boundary layer transition was systematically investigated in the Cryogenic Ludwieg-Tube Gottingen (DNW-KRG). For this investigation the existing two-dimensional wind tunnel model, PaLASTra, which offers a quasi-uniform streamwise pressure gradient, was modified to reduce the size of the flow separation region at its trailing edge. The streamwise temperature distribution and the location of laminar–turbulent transition were measured by means of temperature-sensitive paint (TSP) with a higher accuracy than attained in earlier measurements. It was found that for the modified PaLASTra model the transition Reynolds number (
 $$Re_{\text {tr}}$$
 ) exhibits a linear dependence on the pressure gradient, characterized by $$H_{12}$$
 . Due to this linear relation it was possible to quantify the so-called ‘unit Reynolds number effect’, which is an increase of $$Re_{\text {tr}}$$
 with $$Re_1$$
 . By a systematic variation of M, $$Re_1$$
 and $$H_{12}$$
 in combination with a spectral analysis of freestream disturbances, a stabilizing effect of compressibility on boundary layer transition, as predicted by linear stability theory, was detected (‘Mach number effect’). Furthermore, two expressions were derived which can be used to calculate the transition Reynolds number as a function of the amplitude of total pressure fluctuations, $$Re_1$$
 and $$H_{12}$$
 . To determine critical N-factors, the measured transition locations were correlated with amplification rates, calculated by incompressible and compressible linear stability theory. By taking into account the spectral level of total pressure fluctuations at the frequency of the most amplified Tollmien–Schlichting wave at transition location, the scatter in the determined critical N-factors was reduced. Furthermore, the receptivity coefficients dependence on incidence angle of acoustic waves was used to correct the determined critical N-factors. Thereby, a found dependency of the determined critical N-factors on $$H_{12}$$
 decreased, leading to an average critical N-factor of about 9.5 with a standard deviation of $$\sigma \approx 0.8$$
 .

https://link.springer.com/content/pdf/10.1007/s00348-018-2538-8.pdf

Unit Reynolds number, Mach number and pressure gradient effects on laminar–turbulent transition in two-dimensional boundary layers

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