Development of second mode instability in a Mach 6 flat plate boundary layer with two dimensional roughness Development of second mode instability in a Mach 6 flat plate boundary layer with two dimens[.]
Development of second-mode instability in a Mach flat plate boundary layer with two-dimensional roughness Qing Tang, Yiding Zhu, Xi Chen, and Cunbiao Lee Citation: Phys Fluids 27, 064105 (2015); doi: 10.1063/1.4922389 View online: http://dx.doi.org/10.1063/1.4922389 View Table of Contents: http://aip.scitation.org/toc/phf/27/6 Published by the American Institute of Physics PHYSICS OF FLUIDS 27, 064105 (2015) Development of second-mode instability in a Mach flat plate boundary layer with two-dimensional roughness Qing Tang, Yiding Zhu, Xi Chen, and Cunbiao Leea) State Key Laboratory for Turbulent Research and Complex Systems, College of Engineering, Peking University, Beijing, China and Collaborative Innovation Center of Advanced Aero-Engines, Beijing, China (Received 24 August 2014; accepted 26 May 2015; published online 19 June 2015) Particle image velocimetry, PCB pressure sensors, and planar Rayleigh scattering are combined to study the development of second-mode instability in a Mach flow over a flat plate with two-dimensional roughness To the best of the authors’ knowledge, this is the first time that the instantaneous velocity fields and flow structures of the second-mode instability waves passing through the roughness are shown experimentally A two-dimensional transverse wall blowing is used to generate second-mode instability in the boundary layer and seeding tracer particles The two-dimensional roughness is located upstream of the synchronization point between mode S and mode F The experimental results showed that the amplitude of the second-mode instability will be greatly increased upstream of the roughness Then it damps and recovers quickly in the vicinity downstream of the roughness Further downstream, it acts as no-roughness case, which confirms Fong’s numerical results [K D Fong, X W Wang, and X L Zhong, “Numerical simulation of roughness effect on the stability of a hypersonic boundary layer,” Comput Fluids 96, 350 (2014)] It also has been observed that the strength of the amplification and damping effect depends on the height of the roughness C 2015 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4922389] I INTRODUCTION Hypersonic boundary layer transition remains a popular issue with the rapid development of aeronautics and astronautics For hypersonic boundary layers, there has recently been a strong interest in transition affected by surface roughness Schneider1 mentioned that the mechanism of roughness-induced hypersonic boundary layer transition is very complex The transition process will be influenced by the roughness-induced wake,2–4 the crossflow,5 Görtler instability,6 transient growth7,8 caused by streamwise vorticity, and the interaction between the roughness and boundary layer instability waves or freestream acoustic waves Schneider’s review focused on the boundary layer transition caused by the roughnessinduced unstable wake However, the interaction between the roughness and the boundary layer instability waves, especially the second-mode instability waves, has not received much attention Since Mack9,10 defined second-mode instability in the hypersonic boundary layer, this kind of instability was believed to be one of the dominant instabilities in the hypersonic boundary layer and discovered experimentally by Kendall,11 Demetriades,12 Stetson and Kimmel,13–15 and Casper et al.16,17 The second-mode instability has a large amplification rate in the hypersonic boundary layer Its interaction with roughness should not be ignored in roughness-induced hypersonic boundary layer transition.18 Fedorov 19 studied the receptivity of the hypersonic boundary layer to acoustic disturbance scattered by two-dimensional roughness theoretically and found that the second mode excitation a) Electronic mail: cblee@mech.pku.edu.cn 1070-6631/2015/27(6)/064105/20 27, 064105-1 © Author(s) 2015 064105-2 Tang et al Phys Fluids 27, 064105 (2015) over a two-dimensional local hump strongly depended on the distance between the hump center and the branch point of different instability modes Duan et al.20 made a numerical simulation about a Mach 5.92 flat plate boundary layer under the combined effect of arbitrary finite-height surface roughness and found that the roughness element will stabilize or destabilize the second-mode instability depending on its locating upstream or downstream the synchronization point between mode S and mode F Marxen et al.21 conducted a direct numerical simulation (DNS) study of the disturbance amplification in a Mach 4.8 flat plate boundary layer with a localized two-dimensional roughness element and found that the roughness element considerably alters the instability of the boundary layer, leading to increased amplification or damping of the modal wave depending on its frequency range Most recently, Fong22,23 numerically studied the two-dimensional roughness effect on the second mode instability waves He mentioned that if a roughness element is placed downstream of the synchronization point, the perturbations that have higher frequencies would be damped It has also been found that if the roughness height is less than the local boundary layer thickness, large roughness height results in stronger amplification or stronger damping, depending on the roughness location Heitmann and Radespiel24 also simulated the evolution of a wave packet on a cone with roughness However, there is still a lack of experimental results about the development of the frequency and amplitude of the second-mode instability waves upstream and downstream of the roughness Fujii25 studied the effects of wavy-wall roughness on hypersonic boundary layers on a five-degree half-angle sharp cone at Mach 7.1 He found that the wavy-wall roughness, located well upstream of the breakdown region, could delay the transition under some conditions However, he did not focus on the roughness’ effect on the second-mode instability waves upstream and downstream The present study is conducted in the 120 mm Mach quiet tunnel at Peking University A two-dimensional transverse wall blowing is used to generate second-mode instability waves in the flat plate boundary layer A two-dimensional roughness element is installed at 125 mm from the leading edge of the flat plate, upstream of the synchronization point between mode S and mode F PCB pressure sensors are utilized to measure the second-mode instability waves upstream and downstream of the roughness element Particle image velocimetry (PIV) is used to show the flow structures upstream and downstream of the roughness, compared to the PCB results Planar Rayleigh scattering (PRS) flow visualization is also taken as a supplement to the PIV and PCB results II EXPERIMENTAL FACILITIES The experimental parameters are listed in Table I A Wind tunnel The wind tunnel operates in a pressure-vacuum blow down infrastructure with a typical on-condition test time of 30 s To avoid air liquefaction, the flow is heated to a nominal stagnation temperature of T0 = 433 K The stagnation pressure is held constant, and the unit Reynolds number Reunit = ρ∞U∞/µ changes less than 3% in a running time The tunnel operates at quiet and noisy conditions with the bleed valve open and closed, respectively The freestream flow density ρ∞ = 0.021 kg/m3 and the viscosity coefficient = 0.34 ì 105 Ns/m2 can be calculated by TABLE I The experimental parameters of the tunnel at PKU D T0 P0 Reunit U∞ RMS p ′/p (noisy) 120 mm 433 K 0.5 MPa 5.40 × 106 1/m 873 m/s 2.5% 064105-3 Tang et al Phys Fluids 27, 064105 (2015) FIG The flat plate model with PCB pressure sensors mounted at 90 mm–180 mm downstream of the leading edge The two-dimensional transverse wall blowing and two-dimensional roughness are also shown Sutherland’s law in the form µ = µ 0( T 3/2 T0 + Ts ) , T0 T + Ts (1) where for air, à0 = 1.72 ì 105 Ns/m2 and Ts = 110.4 K B Test model The flat plate model is shown in Fig A 350 mm long flat plate was installed in the test section, mm off the axis of the nozzle The flat plate is 90 mm wide at the leading edge and 60 mm wide at the trailing edge The leading edge was sharp and beveled with an angle of 15◦ The siding edges were also sharpened and beveled at an angle of 45◦ Wall blowing procedure is utilized to enhance the second-mode instability waves The wallblowing slot is two-dimensional (3 mm × 40 mm) and 50 mm downstream of the leading edge The blowing rate is set at L/min Since the wall blowing will disturb the boundary layer more or less, the experimental area should be chosen as far downstream as possible On the other hand, limited by the nozzle exit, the oblique shock wave induced by the leading edge of the flat plate is reflected by the boundary layer on the nozzle and hits on the flat plate at about 200 mm downstream of the leading edge of the flat plate Therefore, our experimental zone is set from x = 90 mm to x = 180 mm on the flat plate model C Roughness location Limited by the experimental zone, three two-dimensional roughness elements with different heights (0.5 mm, 1.0 mm, 1.5 mm) (H) × mm (L) × 60 mm (W) are mounted at 125 mm from the leading edge on the flat plate, respectively Here, the development of the second mode instability waves can easily be detected upstream and downstream of the roughness Since the relationship between the roughness location and the synchronization point of mode S and mode F plays an important role in the developments of mode S,22 we should first define the synchronization point in our case The synchronization location in the x coordinate can be calculated by22 xs = (ω s /F)2 Reunit (2) The synchronization point has a constant value of non-dimensional circular frequency ω s = 0.114, which is calculated by Linear Stability Theory (LST) and shown in Fig The dimensionless 064105-4 Tang et al Phys Fluids 27, 064105 (2015) FIG Distributions of phase velocities a and the dimensionless circular frequency ω for boundary-layer waves The mode S synchronizes with mode F at the point where ω s = 0.114 and a s = 0.927 frequency F = 1.2 × 10−4 is defined as F= 2π f ν , U∞2 (3) where the kinematic viscosity = = 1.62 ì 104 m2/s and the typical frequency of the second mode instability f = 90 kHz, which will be shown in Eq (5) Thus, we have the synchronization location in the x coordinate x s = 0.167 m This means that the roughness in our case is placed upstream of the synchronization point between mode S and mode F D Experimental methods Nine PCB pressure sensors are installed from 90 mm to 180 mm downstream of the leading edge of the flat plate as shown in Fig The PCB132A31 pressure sensor is a very high frequency piezoelectric sensor with a resolution of approximately Pa The resonant frequency of the sensor is above MHz, and the sensor output is high-pass filtered at 11 kHz and, hence, suitable to measure the high-frequency instabilities in the hypersonic boundary layer The output from the PCB pressure sensors passes through a PCB 482C01 signal conditioner A DH5939 data acquisition system is used for data acquisition It has a bandwidth of 600 kHz and a 14-bits vertical resolution The sampling rate is MHz The detailed parameters of the PCB pressure sensor are mentioned by Fujii.25 As a supplement to the PCB results, the PRS is used to visualize the flow structure of the hypersonic flat plate boundary layer Carbon dioxide is injected into the tunnel from upstream of the electric heater The mass injection rate of the carbon dioxide is no more than 5% of the freestream flow Owing to the low static temperature in the test section of the tunnel (less than 70 K), the condensation process converts it into dry cold particles Given that the diameters of the particles are much less than the wavelength of the laser, this is a kind of Rayleigh scattering On the other hand, the temperature near the wall is high enough that the solid-state carbon dioxide becomes a gas state The line separating the black and white parts in the photograph is the condensation line, as shown in Figs 19 and 20 The disturbance in the boundary layer develops, and the black and white parts mix together to visualize the flow structure This technique had been used by Smits’ group26–28 in many kinds of hypersonic boundary layers PIV is used to measure the near-wall boundary layer velocity field on the flat plate In our experiment, tracer particles could hardly get into the laminar and transitional hypersonic boundary layers because of the Saffman force29 caused by the large shear stress near the wall The laminar and transitional hypersonic boundary layers also not have enough mass convection as fully developed turbulent boundary layer The transverse wall blowing used to inject tracer particles into the boundary layer could partly solve this problem However, it still faces many other challenges in hypersonic boundary layer measurement, such as large movement of the tracer particles, large velocity gradient in the thin boundary layer, and intense scattering near the roughness and the wall 064105-5 Tang et al Phys Fluids 27, 064105 (2015) FIG Schematic diagram of the experiment The preprocessing and post-processing of the PIV images are critical to the final velocity field results, since the experiment is in a high-speed and high shear flow condition A great deal of work has been performed to improve the hypersonic boundary PIV results, such as the prior calculation of the initial velocity field, wall mask, roughness mask, wall scattering mask, wall weight arrangement, and flow structure smoothing This improved PIV method had been successfully utilized to measure the velocity profile of a Mach fully developed turbulent boundary layer The RMS uncertainty is about 0.02-0.05, and the mean bias of the PIV method is about 0.01-0.015, depending on the displacement gradient.30 The flow was seeded with particles of 90 nm in diameter with transverse wall blowing A Nd:Yag dual-head laser of 400 mJ provided the illumination and a 12 bits TSI imager intense recorded the scattered light intensities A 200 mm lens is used to take photographs at three different areas at streamwise direction, each of which is time-independent and has an image size of 20 mm × 16 mm, with a conversion factor of pixel per 1.5625 µm in the image plane The semantics of the PIV experiment are shown in Fig III RESULTS AND DISCUSSION A Effect of wall blowing First, the effect of transverse wall blowing to the boundary layer is investigated Since the amplitude of the second-mode instability waves is not so large, a linear y-axis is used to show the amplitude development of the second-mode instability waves more clearly The total pressure is 0.5 MPa, and the unit Reynolds number Reunit = 5.40 × 106 1/m Fig is the power spectral density (PSD) of the wall pressure fluctuation, without wall blowing measured by PCB pressure sensors We can see that the typical frequency band of the pressure fluctuation is between 80 kHz and 120 kHz, which is in the frequency band of the second-mode instability waves on the flat plate.11 The fluctuation is also amplified with streamwise distance Hence, it can be confirmed that this kind of fluctuation is caused by the second-mode instability waves and its peak frequency is about 105 kHz From Fig 5, we can see that the typical frequency band of the pressure fluctuation is between 60 kHz and 100 kHz, which is also in the frequency band of the second-mode instability waves on the flat plate The typical frequency band of the fluctuation in wall blowing case is lower, since the 064105-6 Tang et al Phys Fluids 27, 064105 (2015) FIG PSD of the surface pressure fluctuation, without wall blowing, measured by PCB pressure sensors boundary layer is thicker Its peak frequency is 95 kHz and the fluctuation amplitude in this frequency band is about an order of magnitude larger than the fluctuation amplitude in the no-blowing case in Fig Hence, it can be confirmed that transverse wall blowing could enhance the amplitude of the second mode instability waves in hypersonic flat plate, which is also mentioned by Johnson.31 Ghaffari32 made a simulation which agreed well with Johnson’s31 result and considered that the wall blowing will bring extra mass flow and thicken the boundary layer, which will, in turn, decrease the frequency of the second-mode instability waves and enhance its amplitude The separation caused by wall blowing will also decrease the viscosity of the flow and make the disturbance unstable as mentioned by Pagella.33,34 Balakumar’s result35 also confirms that the adverse pressure gradient will amplify the first and second mode disturbances In this section, the wall blowing was proven to be able to enhance the amplitude of the second mode instability waves by our experimental result This method was used to make the second mode instability on the flat plate more clear to be measured B Flat plate case Fig is the average PIV velocity field of the hypersonic boundary layer The results of 40 pairs of recordings were averaged to give the mean result The total pressure is 0.5 MPa, and the injection FIG PSD of the surface pressure fluctuation, with wall blowing, measured by PCB pressure sensors 064105-7 Tang et al Phys Fluids 27, 064105 (2015) FIG Average PIV velocity field of the hypersonic boundary layer at 90 mm–150 mm downstream of the leading edge of the flat plate rate of the wall blowing is L/min The boundary layer thickness is δ ≈ mm from about 90 mm to 150 mm downstream of the leading edge of the flat plate We extract the streamwise velocity in different x locations and compare them with the Blasius velocity profile, shown in Fig Since the boundary layer is disturbed by wall blowing and is no longer a naturally developed laminar boundary layer, an x = 150 mm shift has been done to fit the PIV results with the Blasius profile The velocity defects in the lower region and plumps in the upper region of the boundary layer, which is shown in Fig 7, may be caused by the transverse wall blowing The mean bias errors of the PIV results are less than 10% of the freestream velocity, which is shown in Fig Fig shows the surface pressure fluctuation of a flat plate measured by PCB pressure sensors The pressure data are band-pass filtered from 60 kHz to 100 kHz It can be seen that the amplitude of the second-mode instability waves increases along the streamwise direction The speed of the second-mode instability waves could be calculated by two-point correlation of the surface pressure fluctuation measured by PCB pressure sensors, which is shown in Fig 10 For the two points, distance is 20 mm, the correlation time is 0.0276 ms, and the speed of the second-mode instability waves is u = x/t = 20 mm/0.0276 ms ≈ 724 m/s (4) Since the wavelength of the second-mode instability is about twice the boundary layer thickness,11,13 the frequency of the second-mode instability waves is f = u/L ≈ u/2δ ≈ 90 kHz, (5) which is also in the frequency band of the second-mode instability measured by the PCB pressure sensors Fig 11 is the instantaneous PIV velocity field of the hypersonic boundary layer, with the second-mode instability waves appearing at 90 mm–150 mm downstream of the leading edge of the flat plate The average wavelength is about mm-8 mm, and it can also be discovered that the shape of the waves is not so regular, including random disturbances of different frequencies This is reasonable since the amplitude of the second-mode instability waves in the boundary layer of the flat plate is not as large as the cone and the disturbances of other frequencies are also effective FIG Comparison of the streamwise velocity profile between the mean results of PIV and the Blasius equation The PIV result in 100 mm, 120 mm, and 140 mm has been shifted to 250 mm, 270 mm, and 290 mm, respectively 064105-8 Tang et al Phys Fluids 27, 064105 (2015) FIG The mean bias errors of the PIV results to the Blasius profile C Effect of two-dimensional roughness Three kinds of two-dimensional roughness elements with different heights of 0.5 mm, 1.0 mm, and 1.5 mm are installed 125 mm downstream of the leading edge of the flat plate For the boundary layer, thickness δ ≈ mm, and the heights of the roughness that we chose are about 0.125δ, 0.25δ, and 0.375δ, respectively The pressure data are band-pass filtered from 60 kHz to 100 kHz FIG Surface pressure fluctuation measured by PCB pressure sensors mounted in the streamwise direction The signal is band-passed between 60 kHz and 100 kHz There is no roughness 064105-9 Tang et al Phys Fluids 27, 064105 (2015) FIG 10 The two-point correlation of the surface pressure fluctuation between x = 110 mm and x = 130 mm 0.375δ roughness Fig 12 shows the surface pressure fluctuations upstream and downstream of the 0.375δ roughness It can be seen in Fig 12 that at mm and 15 mm upstream of the roughness, the amplitude of the second-mode instability is greatly enhanced, and then it damps and recovers at near downstream of the roughness Further downstream, the second-mode instability waves redevelop in the no-roughness case Fig 13 is the comparison of the normalized spatial evolution of the wall pressure fluctuation between experimental and numerical results As mentioned above, the roughness at x = 125 mm is located upstream of the synchronization point between mode S and mode F The simulation result used for comparison is provided by Fong.22 Although having different shapes of the roughness, boundary layer thickness, and initial disturbances, the numerical and experimental results fit well, after necessary normalization It can be seen in Fig 13 that upstream of the roughness (x = 105 mm–110 mm at Fong’s computational fluid dynamics (CFD) results and x = 120 mm in our results), the amplitudes of the second mode instability waves increase about 2.5-3 times as the no-roughness case In the region behind and close to the roughness, the amplitude of the second mode instability waves is found damped and recovered quickly at x = 115-122 mm in the numerical simulation However, this phenomenon is not so clear at x = 130-140 mm in the experimental results, which is caused by the band-pass filter of the result (the signal is band-passed between 60 kHz and 100 kHz, but the typical frequency of the second mode instability is changing upstream and downstream of the roughness) In fact, the damping and recovering of the second mode instability physically exist and can be seen more clearly in Fig 24 (high-pass filtered at 11 kHz) Further downstream of the roughness, the amplitude of the second mode instability waves is found to develop similarly to the no-roughness case in our experimental results This confirms Fong’s22 numerical simulation result that when the roughness is located upstream the synchronization point between mode S and mode F, the amplitude of the second mode instability waves downstream of the roughness is amplified similarly to the no-roughness case FIG 11 Instantaneous PIV velocity field of the hypersonic boundary layer at 90 mm–150 mm downstream of the leading edge of the flat plate The vectors here indicate the velocity directions 064105-10 Tang et al Phys Fluids 27, 064105 (2015) FIG 12 Surface pressure fluctuation measured by PCB pressure sensors mounted in the streamwise direction The signal is band-passed between 60 kHz and 100 kHz The 0.375δ roughness is mounted at 125 mm downstream of the leading edge of the flat plate The instantaneous flow fields upstream and downstream of the 0.375δ roughness measured by PIV are shown in Fig 14 Here, we can see the velocity contour and streamline from 90 mm to 150 mm downstream of the leading edge of the flat plate Fig 14 shows the instantaneous velocity field, and the three parts in Fig 14 are time-independent The flow separates upstream of the roughness, and the boundary layer is lifted outside of the FIG 13 Normalized RMS of the surface pressure fluctuation at the streamwise direction p ′ is the RMS of the surface pressure fluctuation, and p 90 is the RMS of the surface pressure fluctuation at x = 90 mm The pressure fluctuation in the numerical simulation results by Fong is also normalized by the pressure fluctuation at x = 90 mm The red dashed line indicates the experimental result with 0.375δ roughness, and the thick one indicates the relevant roughness location The blue line indicates the numerical simulation result22 with 0.375δ roughness, and the thick one indicates the relevant roughness location The black dashed line and the grey line indicate the no-roughness cases in the experimental and numerical results, respectively 064105-11 Tang et al Phys Fluids 27, 064105 (2015) FIG 14 Instantaneous flow fields upstream and downstream of the 0.375δ roughness separation zone When passing through the roughness element, the separated boundary layer reattaches onto the wall The detailed instantaneous velocity profile around the separation zone is extracted from Fig 14 and is shown in Fig 15 Fig 15 shows that the flow separates at about 21 mm upstream of the roughness, which is 14 times the height of the roughness In Marxen’s DNS results,21 the length of the separation zones upstream and downstream of the roughness is about 13 times and six times the height of the roughness, respectively, which is similar to our measurement The PIV results also fit well with Balakumar’s36 CFD results 0.25δ roughness Fig 16 is the surface pressure fluctuations upstream and downstream of the 0.25δ roughness The development of the surface pressure fluctuations upstream and downstream of the 0.25δ roughness is nearly the same as the 0.375δ roughness case It can be seen in Fig 16 that at mm and 15 mm upstream of the roughness, the amplitude of the second-mode instability is greatly enhanced, and then it damps and recovers at near downstream of the roughness Further downstream, the second-mode instability waves redevelop as the no-roughness case It can be seen in Fig 17 that upstream of the roughness (x = 105 mm–110 mm at Fong’s CFD results and x = 120 mm in our results), the amplitudes of the second mode instability waves increase about 2.5 times those of the no-roughness case In the region behind and close to the roughness, the amplitude of the second mode instability waves is found damped and recovered quickly at x = 115-120 mm in the numerical simulation Based on the same previously mentioned reason, the damping of the second mode instability at x = 130-140 mm in our experiment is not so strong Further downstream of the roughness, the amplitude of the second mode instability waves is found to develop similarly to the no-roughness case, consistent with Fong’s results.22 The instantaneous flow fields upstream and downstream of the 0.25δ roughness measured by PIV are shown in Fig 18 In this picture of the velocity field, the boundary layer separates a short distance upstream of the roughness It then reattaches after passing around the roughness downstream The shape of the second-mode instability waves is not so regular downstream of the FIG 15 Instantaneous velocity profile upstream the 0.375δ roughness 064105-12 Tang et al Phys Fluids 27, 064105 (2015) FIG 16 Surface pressure fluctuation measured by PCB pressure sensors mounted in the streamwise direction The signal is band-passed between 60 kHz and 100 kHz The 0.25δ roughness is mounted at 125 mm downstream of the leading edge of the flat plate roughness The large and clear separation zone existing in the 0.375δ roughness case is not found in the 0.25δ roughness case In order to obtain a clear picture of the second-mode instability waves passing through the roughness, a PRS flow visualization is used to visualize the flow structures as a supplement to the FIG 17 Normalized RMS of the surface pressure fluctuation at the streamwise direction p ′ is the RMS of the surface pressure fluctuation, and p 90 is the RMS of the surface pressure fluctuation at x = 90 mm The pressure fluctuation in the numerical simulation results by Fong is also normalized by the pressure fluctuation at x = 90 mm The red dashed line indicates the experimental result with 0.25δ roughness, and the thick one indicates the relevant roughness location The blue line indicates the numerical simulation result22 with 0.25δ roughness, and the thick one indicates the relevant roughness location The black dashed line and the grey line indicate the no-roughness cases in the experimental and numerical results, respectively 064105-13 Tang et al Phys Fluids 27, 064105 (2015) FIG 18 Instantaneous flow fields upstream and downstream of the 0.25δ roughness PIV results It is commonly used in supersonic and hypersonic flow experiments,26–28 based on the condensation of carbon-dioxide and its light scattering It has successfully shown the regular second-mode instability waves in a Mach conical boundary layer.37 Carbon dioxide is injected into the tunnel upstream of the electric heater Its mass flow rate is about 5% of the freestream flow From Fig 19, it can be seen that the boundary layer is lifted a little upstream of the 0.25δ roughness The second-mode instability waves’ shape remains unchanged when passing over the roughness, which fits well with our PIV results Fig 20 shows the flow structures downstream of the 0.25δ roughness The shape of the second-mode instability waves becomes irregular downstream of the roughness, which can also be seen in Fig 18 Many flow structures with different sizes and shapes developed downstream of the roughness These phenomena may not be caused by the dispersion of the instability waves because Fig 24 clearly showed that the second mode instability waves redeveloped downstream of the roughness, without any distortion A possible mechanism of the irregularity of the second mode instability waves was given by Marxen et al.,38 in which the amplification may be reduced due to nonlinear effects 0.125δ roughness Fig 21 shows the surface pressure fluctuations upstream and downstream of the 0.125δ roughness It can be seen in Fig 21 that, upstream and downstream of the roughness, the amplitude of the second-mode instability waves is not much influenced by the roughness It developed nearly the same as the flat plate case, which confirms Fong’s CFD results22 that roughness of a lower height has a smaller effective zone and a weaker effective strength in the hypersonic boundary layer FIG 19 Instantaneous PRS flow visualizations upstream and downstream of the 0.25δ roughness 064105-14 Tang et al Phys Fluids 27, 064105 (2015) FIG 20 Instantaneous PRS flow visualization downstream of the 0.25δ roughness Although there is no corresponding numerical simulation result about the 0.125δ roughness, Fig 22 also demonstrates experimentally that the amplitude of the second mode instability waves develops nearly the same as the no-roughness case The instantaneous flow fields upstream and downstream of the 0.125δ roughness measured by PIV are shown in Fig 23 FIG 21 Surface pressure fluctuation measured by PCB pressure sensors mounted in the streamwise direction The signal is band-passed between 60 kHz and 100 kHz The 0.125δ roughness is mounted at 125 mm downstream of the leading edge of the flat plate 064105-15 Tang et al Phys Fluids 27, 064105 (2015) FIG 22 Normalized RMS of the surface pressure fluctuation at the streamwise direction p ′ is the RMS of the surface pressure fluctuation, and p 90 is the RMS of the surface pressure fluctuation at x = 90 mm The red dashed line indicates the experimental result with 0.125δ roughness, and the thick one indicates the relevant roughness location The black dashed line and the grey line indicate the no-roughness cases in the experimental and numerical results, respectively In this picture of the velocity field, the second-mode instability waves are passing across the roughness without any change The thickness of the boundary layer is nearly the same when passing through the roughness The PIV results are consistent with the PCB results Summary In this section, a summary of the three cases of roughness with different heights and the no-roughness case is made The summary is mainly from the wall pressure fluctuation data measured by PCB pressure sensors The development of the second mode instability waves was shown in Fig 24, which is the PSD of the wall pressure fluctuation measured by PCB pressure sensors The changing of the second-mode instability waves’ frequency and amplitude was shown more clearly Upstream of the 0.25δ roughness, the amplitude of the second-mode instability waves increased, and its frequency decreased relatively, which is shown in Fig 24, at x = 120 mm In the 0.375δ roughness case, the amplitude of the second-mode instability waves is about 1.5 times larger than the 0.25δ roughness case upstream of the roughness at x = 120 mm, and its frequency becomes smaller than the 0.25δ roughness case, which is also mentioned by Marxen.21 As shown in Fig 14, the roughness could separate and thicken the boundary layer upstream This will cause a higher Reθ and a lower viscosity,34 which could make the second mode instability waves more unstable This mechanism is similar to the amplitude enhancement of the second mode instability in the wall blowing case Mack10 mentioned that viscosity could stabilize the second mode instability, “The shapes of the neutral-stability curves, both before and after merger, are such as to suggest that viscosity is only stabilizing for all higher modes, and this is confirmed for the 2D second mode .” Pagella33 also indicated that “first, decreasing viscosity effects due to the displacement of the boundary layer away from the wall are responsible for a certain rise in instability, because at higher Mach numbers viscosity has a known stabilizing effect Also, local relative supersonic zones become thicker within the region where the boundary layer thickens This is known to promote second-mode instability.” FIG 23 Instantaneous flow fields upstream and downstream of the 0.125δ roughness 064105-16 Tang et al Phys Fluids 27, 064105 (2015) FIG 24 PSD of the surface pressure fluctuation at the streamwise direction of different roughness cases The black line indicates the no-roughness case; blue line indicates the 0.125δ roughness case; red line indicates the 0.25δ roughness case; green line indicates the 0.375δ roughness case In the region behind and close to the 0.25δ and 0.375δ roughness, the amplitude of the second-mode instability waves is strongly damped and then recovered, as shown in Fig 24 The pressure fluctuation measured by PCB pressure sensors is very weak on the wall near downstream of the roughness The second mode instability could also not be detected in its typical frequency range Fig 25 also shows that the velocity fluctuation in the separation zone downstream of the roughness is much smaller than in the upper boundary layer This indicated that the velocity FIG 25 RMS of the velocity field of the 0.25δ roughness case, calculated by 40 frames of instantaneous PIV results The dashed lines indicate the sonic line and vortex sheet of the flow downstream of the roughness 064105-17 Tang et al Phys Fluids 27, 064105 (2015) FIG 26 A schematic diagram of the flow structure downstream of the roughness fluctuation cannot penetrate the dashed line, as shown in Fig 25 The dashed line between the separation zone downstream of the roughness and the upper boundary layer acts as a vortex sheet Based on our PCB and PIV results, it appears that the second mode instability wave could hardly penetrate the vortex sheet The schematic diagram of the flow structure downstream of the roughness is shown in Fig 26 Since the second mode instability waves are trapped acoustic waves between sonic line and the wall physically,39 the problem here is simplified as a sound wave passing the vortex sheet between two regions with relative motion The shear layer between the two regions is approximated by a vertex sheet, as we assume that its thickness is much smaller than the wavelength of the second mode instability waves (λ ≈ mm) The pressure distribution in the wall normal direction and the influence of the expansion fan and shock wave in the boundary layer are not taken into account Figure 27 is the average velocity field in the vicinity of the 0.25δ roughness, which is averaged by 40 frames of instantaneous PIV results The upper dashed line indicates the sonic line: y = ya , U( ya ) = c − a( ya ) ≈ 650 m/s, where c = 0.92U∞ ≈ 800 m/s is the disturbance phase speed provided by LST, U( y) is the mean-flow profile, and a ≈ 150 m/s is the local speed of sound The lower dashed line indicates the vortex sheet, which has a velocity v ≈ 200 m/s The mean velocity of the lower region is approximately 100 m/s Based on Miles’ theory,40 the relative amplitude of the reflected wave into the high speed region can be given by sin 2α − sin β , sin 2α + sin β c2 c1 sin β = , sin α = , v − V2 v − V1 R= (6) (7) FIG 27 The average velocity field of the 0.25δ roughness case The dashed lines indicate the sonic line and vortex sheet of the flow downstream of the roughness 064105-18 Tang et al Phys Fluids 27, 064105 (2015) TABLE II The peak frequency of the second-mode instability waves No roughness (kHz) 0.125δ roughness (kHz) 0.25δ roughness (kHz) 0.375δ roughness (kHz) 90 mm 110 mm 120 mm 130 mm 140 mm 150 mm 160 mm 170 mm 180 mm 95 90 90 95 95 95 95 95 95 95 90 90 95 95 95 95 95 100 95 90 70 80 80 80 80 95 85 65 85 85 85 85 c2 sin β = 2( ) v − V2 ( v − V2 ) − 1, c2 (8) where α is the reflection angle and β is the refraction angle, v is the velocity of the vortex sheet, and V1,V2 and c1, c2 is the average velocity and sound speed of the two regions, respectively In our case, we assume that the temperature of the separation region remains a constant equal to the temperature of the wall Thus, the sound speed in the separation region c2 ≈ 340 m/s As shown in Fig 27, the average velocity of the separation region V2 ≈ 100 m/s, and the speed of the vortex sheet v ≈ 200 m/s, then we have v < V2 + c2 From Eq (8), sin β is a purely imaginary quantity; the total reflection will occur (|R| = 1) at any wave injection angles This means that the sound waves could not propagate in the low speed separation region However, it just a heuristic discussion with many simplifications to the complex flow field More rigorous analysis considering real flow conditions needs to be done in the future work After reattachment, the disturbance evolution returns to a flat plate case behavior, which is similar to Duan’s20 and Fong’s22 results This result proved that when the 2D roughness is located upstream the synchronization point between mode S and mode F, the development of the second mode instability waves downstream of the roughness is similar to the no-roughness case The peak frequency of the second-mode instability waves upstream and downstream of the roughness with different heights is listed in Table II The peak frequency of the second mode insatiability waves was found to be greatly decreased upstream of the 0.25δ and 0.375δ roughness, and the second-mode instability waves damped not faraway downstream of the 0.25δ and 0.375δ roughness This is because the boundary layer upstream of the roughness is thickened by the separation, which will increase the wavelength of the second-mode instability and decrease its frequency As Malik and Anderson41 described, anything that thins the boundary layer decreases the wavelength and thus increases the frequency and the converse is also true In Table II, it can be seen that the frequency of the instability waves is nearly the same between the 0.125δ roughness case and the no-roughness case The boundary layer thickness does not change significantly upstream of the roughness, which determines the frequency of the second-mode instability waves From Fig 24, it can also be seen that the wall pressure perturbation is not changed both in terms of amplitude and frequency when passing through the 0.125δ roughness The results indicate that when the roughness height is less than 1/8 of the boundary layer thickness, its influence on the second-mode instability waves is nearly the same as the no-roughness case, which confirms numerical results of Fong22 and Marxen21 that the amplification rates upstream and downstream of the roughness are strongly dependent on the roughness height The small roughness will cause small amplification rate upstream and small damping rate downstream and it also has a small effective zone IV CONCLUSIONS In this paper, PIV velocity measurements and PRS flow visualization are both used to study the interaction between the second-mode instability waves and the two-dimensional roughness in a 064105-19 Tang et al Phys Fluids 27, 064105 (2015) Mach flat plate boundary layer An improved near-wall PIV method is developed and successfully shows the instantaneous flow structures, such as the second-mode instability waves, separation zone upstream of the roughness, and reattachment zone downstream of the roughness PRS visualization is utilized to show the flow structures upstream and downstream of the 0.25δ roughness and is consistent with our PIV results PCB pressure sensors are also used to measure the frequency and amplitude characteristics of the second-mode instability waves upstream and downstream of the roughness elements With the combination of these methods, a complete picture is obtained for the second-mode instability waves’ development in a Mach boundary layer of a flat plate with two-dimensional roughness together with the three main results summarized below First, it is proven that a two-dimensional transverse wall blowing is effective to enhance the amplitude of second-mode instability waves in a hypersonic flat plate boundary layer The frequency of the second-mode instability in the flat plate boundary layer is about 60-100 kHz This finding confirms Johnson’s CFD results31 experimentally Second, the two-dimensional roughness located upstream the synchronization point is found to be effective to the second-mode instability waves locally When the roughness height is about 0.375δ, the frequency of the second-mode instability waves decreases from 85 kHz to 65 kHz upstream of the roughness, and the amplitude of the second-mode instability waves increases at about times that of the no-roughness case When passing through the roughness, the amplitude of the second-mode instability waves damps and recovers quickly in the region behind and close to the roughness A heuristic discussion is utilized to explain this phenomenon Further downstream, the second mode instability waves develop similarly to the no-roughness case, which supports CFD results of Duan20 and Fong.22 Finally, the height of the two-dimensional roughness plays an important role in affecting the amplitude of the second-mode instability waves Larger roughness element causes stronger amplification upstream when it is installed upstream the synchronization point (125 mm downstream of the flat plate leading edge) When the roughness height is 0.375δ, the separation zones upstream and downstream of the roughness are very clear and have similar sizes as Marxen’s result.21 If the roughness height changes to 0.25δ, the separation zones upstream and downstream of the roughness are not so clear Its effect on the development of the wall pressure fluctuations is also weaker than the 0.375δ roughness case When the roughness height becomes even lower to 0.125δ, the boundary layer separation can hardly be seen, and the development of the wall pressure fluctuations is nearly the same as the no-roughness case Few changes in the frequency and amplitude of the second-mode instability waves are discovered upstream and downstream of the roughness ACKNOWLEDGMENTS We would like to acknowledge the support of Xiaowen Wang, Xinliang Li, and Caihong Su for their useful discussions We would also like to express gratitude to Shashank Khurana for his great help in proofreading the paper This work was supported by the National Natural Science Funds of China for Distinguished Young Scholar group under Grant No 10921202 and 11221061 S P Schneider, “Effects of roughness on hypersonic boundary-layer transition,” AIAA Paper 2007-305, 2007 S Tirtey, O Chazot, and L Walpot, “Characterization of hypersonic roughness-induced boundary-layer transition,” Exp Fluids 50, 407 (2011) G Serino, F Pinna, and P Rambaud, “Numerical computations of hypersonic boundary layer roughness induced transition on a flat plate,” AIAA Paper 2012-0568, 2012 P Iyer, S Muppidi, and K Mahesh, “Boundary layer transition in high-speed flows due to roughness,” AIAA Paper 2012-1106, 2012 R H Radeztsky, M S Reibert, and W S Saric, “Effect of isolated micron-sized roughness on transition in swept-wing flows,” AIAA J 37, 1370 (1999) L de Luca, G Cardone, D Chevalerie, and A Fonteneau, “Viscous interaction phenomena in hypersonic wedge flow,” AIAA J 33, 2293 (1995) E Reshotko and A Tumin, “Role of transient growth in roughness-induced transition,” AIAA J 42, 766 (2004) X W Wang and X L Zhong, “Receptivity of a hypersonic flat-plate boundary layer to three-dimensional surface roughness,” J Spacecr Rockets 45, (2008) Boundary-Layer Stability Theory, Part B., edited by L M Mack JPL, Pasadena, California, 1969 Document No 900-277 ...PHYSICS OF FLUIDS 27, 064 105 (2015) Development of second- mode instability in a Mach flat plate boundary layer with two- dimensional roughness Qing Tang, Yiding Zhu, Xi Chen, and Cunbiao Leea) State... wall blowing is used to generate second- mode instability waves in the flat plate boundary layer A two- dimensional roughness element is installed at 125 mm from the leading edge of the flat plate, ... different instability modes Duan et al.20 made a numerical simulation about a Mach 5.92 flat plate boundary layer under the combined effect of arbitrary finite-height surface roughness and found that