1 Introduction

Recently, considerable attention has been focused on the laminar-to-turbulent transition in hypersonic boundary layers due to its crucial importance in hypersonic engineering design and its fundamental science, including aerodynamic heating, entropy generation, and drag^[1-2]. Compared to incompressible and low-Mach-number flows^[3-4], hypersonic transition has not been actively investigated as a result of recently identified complexities. These complexities present a tremendous challenge from an experimental, theoretical, and numerical analysis perspective. In fact, multi-method investigations, where several approaches are used to validate each other, have been exploited as a commonly-used strategy for unknown complex problems^[5-7]. Over the past few years, our understanding of wall-bounded flow dynamics has been greatly advanced by the combination of experimental and numerical analysis. In particular, improvements in experimental techniques might promote similar advancements that resulted in the unexpected discovery of new physics^[8-16]. To this end, accurate instantaneous measurement of the entire near-wall flow field is necessary for both engineering and scientific applications.

A new instability mechanism in the hypersonic transition process is seen in dilatational longitudinal instability waves, such as the second mode which was first identified by Mack^[17] whenever the freestream is supersonic relative to the phase speed of a disturbance. He suggested the terminology "acoustic modes" to describe these modes because they were dominated by acoustic waves. When the waves are reflected in a region between the sonic line and the solid wall, the flow was alternately compressed and expanded due to the boundary conditions at the sonic line. The consequent variations of the density or velocity divergence could then be captured experimentally using techniques such as schlieren. Hofferth et al.^[18] applied a focused schlieren technique on a 0.5 m long, 5° half-angle flared cone in their Mach 6 quiet wind tunnel at Texas A&M University. The density variation caused by the second-mode instability resulted in light intensity variations in the schlieren image, which were acquired by a high-bandwidth photodetector. Using this technique, Hofferth et al.^[18] determined the frequency and amplitude evolution of the second mode. Laurence et al.^[19] investigated a 1.1 m long, 7° half-angle cone in a free-piston-driven reflected-shock wind tunnel under high-entropy conditions. By employing a 1 000 W short-arc Xe lamp together with a camera capable of recording at a frame rate of up to 1 MHz, Laurence et al.^[19] obtained schlieren images of typical "rope-like" structures of the second mode. The intensity variation in the image was indicative of the density variation caused by the compression and expansion processes inherent to the second-mode instability.

Two-dimensional particle image velocimetry (PIV) has demonstrated its superior potential in the investigation of wall-bounded flow problems due to its high spatial resolution capability. The corresponding multi-grid iterative window-correlation scheme was commonly used due to its reliability and accuracy in the investigation of transient flow fields^[20]. However, this method has limited utility when processing near-wall flows. In particular, one of the main problems is the effect of the interface interference on the cross-correlation. If the fluid-solid interface is immersed in a Cartesian window grid, the interrogation windows will inevitably overlap with the interface. Zhu et al.^[21] developed the image parity exchange technique by adding optimal synthetic particles (OSPs) in the solid region to obtain a more accurate prediction for the velocity near the interface. The improved method expands the traditional window deformation iterative multigrid scheme to PIV images with very large displacements that are commonly encountered in measurements in the hypersonic boundary layers. Before the interrogation, stationary artificial particles of uniform size are homogeneously added in the wall region. The initial estimation near the wall is then smoothed by data from both sides of the shear layer to reduce large random uncertainties. Interrogations in the following iterative steps then converge to the correct results to provide accurate predictions for particle tracking velocimetry. The OSP method has been successfully applied in experimental investigations on the initial growth of flow asymmetries over a slender body for revolution at high angles of attack^[22]. The near-wall measurements are even more difficult when the interface is moving and curved, which is typical of turbomachinery, especially in the air. Jia et al.^[23] proposed the adaptive stretching of the image strips method, which made it possible to measure near-wall flows in turbomachines. This is the first successful near-wall measurement over moving interfaces in air, providing both the velocity distribution in the near-wall region and the integral parameters of the boundary layer around a rotating blade. The results emphasize the role of the wake negative jet effect and wake induced high disturbances in affecting the boundary layer development.

In this paper, the application of a combination of PIV and direct numerical simulation (DNS) on the flow structures of a transitional hypersonic boundary layer is investigated. The remainders are organized as follows. The experimental and numerical setups are presented in Section 2. The results and discussion are presented in Section 3, and the main conclusions are highlighted in Section 4.

2 Method description 2.1 Experimental setup

The experiments were performed in a Mach 6 wind tunnel at Peking University, which at present is one of three operational hypersonic quiet wind tunnels in the world. The tunnel is of the open-jet configuration with a nozzle exit diameter of 160 mm. Longitudinal suction can be applied upstream the nozzle throat, and the wind tunnel runs under quiet or noisy conditions with the suction valve open or closed, respectively. In the case of the quiet condition, the turbulence level is below 0.2%. Due to low freestream disturbances and the upper limit of the unit Reynolds number for the quiet condition, a natural transition does not occur up to the end of the present model. Therefore, to obtain a complete picture of the transition process in the present work, we ran the wind tunnel under the noisy condition with the suction valve closed. To avoid liquefying the air, the flow was pre-heated to a nominal stagnation temperature of 433 K. During a typical run time of 30 s, the stagnation pressure maintained a nearly constant value with a variation of less than 3 %. Four flow conditions with Re_unit = 5.4 m^-1, 7.6 m^-1, 9.7 m^-1, and 11.7×10⁶ m^-1 were investigated, where Re_unit is the freestream unit Reynolds number.

The model used in the present study is a flared, instability-enhanced cone, which is schematically shown in Fig. 1. The full length is L = 260 mm. Its geometry consists of a 5° half-angle circular conical profile for the first 100 mm of axial distance, followed by a tangent flare of radius 931 mm until the base of the cone at the 260 mm axial position. The first 50 mm of length, which can be removed and replaced with an arbitrary nose-tip shape, is made of stainless steel. In this work, the tip is sharp with a nominal 50 μm radius, and the rest of the model is made of Bakelite. The origin of the coordinate is located at the tip of the cone, with x being the streamwise coordinate along the cone's surface, y being the coordinate normal to that surface, z being the transverse coordinate normal to the xy-plane, and l being the axial coordinate along the axis of the model.

The cone is installed along the centerline of the nozzle with a zero angle of attack. The tip is 50 mm deep into the nozzle exit. The length of the flow field that is not affected by the reflected Mach waves is more than 400 mm, which is long enough to embed the entire length of the model.

2.2 PIV setup

Due to the rapid developments in charge coupled device (CCD) cameras since the first application of PIV in hypersonic flows, this technique is now commonly used in high-speed flow analysis. The key issues include the selection of seeding particles, their dispersion in the flow, and the analysis of particle image recordings.

To seed the present flow, TiO₂ particles were utilized because of their high-temperature stability and high index of refraction. The nominal diameter of the selected particles was 90 nm. The tracking capability of the particle was assessed by Ref. [20] as the ratio given by

(1)

where τ_p is the relaxation time of the particle as it passes through a velocity step (e.g., an oblique shock wave) from u₁ to u₂, in which u₁ and u₂ are the initial and final velocities during the displacement of a single particle, respectively. The velocity of the particle is given by

(2)

where t is the time. The flow time scale τ_f is the slipping time of the flow, defined as

(3)

where du is the velocity spatial variation, and dl is the distance for that variation.

In the present work, τ_p is assessed experimentally to be 2 μs. For a hypersonic boundary layer with the thickness δ, dl is characterized by the wavelength of the second-mode wave λ, and du is no more than 5% of the freestream velocity U_∞. Thus,

In view of the experimental conditions,

Therefore, τ_f≃58.5 μs, and τ≃0.034. The injection of PIV particles slightly dampened the amplitude of the high-frequency second mode. Therefore, the velocity fluctuations below the sonic line are slightly decreased.

The particles are illuminated by a double-cavity Nd:YAG laser from continuum, generating 6 ns duration light pulses at a wavelength of 532 nm with a maximum energy of 2 J per pulse. The laser sheet is projected vertically from the top window of the test section and aligned with the cone's top centerline. The light-sheet's thickness is 1 mm with a width of 100 mm. The time delay of the laser pulses is set as 1 μs, and the sample rate is 2.5 Hz. The light scattered by the particles is viewed by a PCO SensiCam QE CCD camera from a side window, which is equipped with a Nikkor Micro 200 mm lens. The field of view is 34×26 mm².

To conduct window-correlation of particle images, at least 10 particles are required in each interrogation window. However, this is not always satisfied in laminar boundary layers. Furthermore, the low signal-to-noise ratio of the particle images can decrease the Q-value of the correlation peak, thereby either submerging this signal in the noise or yielding incorrect peaks. Therefore, only the images that satisfy both standards are selected.

2.3 Data reduction

In an image set, the interface position can be extracted by using an iterate Radon transformation^[23], based on the equation

(4)

where δ is the Dirac function, and

(5)

in which θ_i is the incidence angle of the line segment.

The analysis of PIV images of a moving body requires regularized coordinates for uniform application of the very effective hierarchical schemes for multi-resolution evaluation. The wall-normal (n) and wall-parallel (s) directions are chosen as the axes of the regularized coordinate system. The image transformation converts the near-wall curved image strips in the physical domain (x, y) to regularized coordinates in the transformed domain (s, n), as shown in Fig. 2.

The sample points are initially created on the wall boundary (n=0) with their distance defined as

(6)

Then, the tangential distribution of the intensity I'(s, 0) at n=0 is interpolated by the following equation:

(7)

where d_k-1 ≤ s < d_k. The normal distribution of the intensity at each location s_k can be interpolated by its neighboring samples (x_k, y_k),

(8)

The relation depicted above can be extended into a two-dimensional condition as follows:

(9)

where

(10)

The image transformation can be depicted as a linear Jacob matrix J as

(11)

where

(12)

To solve the problem of the effect of a very large in-plane displacement and its gradient on the PIV images, which is commonly encountered in hypersonic boundary layers, Zhu et al.^[21] developed an improved image-preprocessing method. PIV images are initially rotated to make the wall boundary nearly horizontal. The evaluations start in 256×256 pixels coarse samples and then 128×128 pixels medium samples to capture the mean displacement fields of the time series. With the mean results as a reference, instantaneous recordings are then evaluated with 64×64 pixels samples to resolve the fine structures in the boundary layer. Smaller samples are unavailable because of the low particle density near the wall. Vectors are then post-processed using a global histogram filter and a Q-ratio of 2. Vectors can then be replaced if a second correlation peak proves to be suitable.

The calculated velocity fields can be reversely transformed to the physical domain by

(13)

The relationship between the velocities in the two domains can be obtained by the inter-coordinate differential relations given by

(14)

where (u, v) and (U, V) represent the velocity vectors in the transformed domain and the real domain, respectively. Two stretch factors are defined as

(15)

with a rotation angle given by

(16)

2.4 Numerical setup

DNS has been performed on the supercomputer Tianhe-2 in the Guangzhou Supercomputing Center. The basic laminar (steady) flow is initially established including the bow shock without any perturbation. The flow is then perturbed to obtain time-dependent simulations of the transitional flow. The code Hoam-OpenCFD is used in these simulations. In this code, the Navier-Stokes equations are solved numerically and presented as follows:

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

A high-order finite-difference method is applied, and the following methods are included.

(Ⅰ) Convection terms are split using Steger-Warming splitting, and are discretized with a seventh-order weighted essentially non-oscillatory (WENO) scheme.

(Ⅱ) An eighth-order central finite-difference scheme is used for the viscous terms.

(Ⅲ) A third-order total-variation-diminishing type of Runge-Kutta method is used for time advance.

The computational domain of the unsteady (transition) simulations in the circumferential direction spans from the 0° meridian plane to the 45° plane. The mesh number (streamwise × wall normal × circumference) is 4 000×200×500, as shown in Fig. 3. Both the curvature effects and external-pressure-gradient effects are taken into account. The boundary conditions for the simulations of the unsteady flow are as follows.

(ⅰ) For the inflow boundary and upper boundary, time-independent conditions are obtained from the steady-flow simulations. The upper boundary is perturbed by random velocity noise with the maximum velocity fluctuations of 1%. Meanwhile, eighteen modes, including first- and second-mode waves, are seeded at the inflow boundary in the form of

(30)

The amplitude ϵ, measured in terms of the peak streamwise velocity fluctuations across the boundary layer, is equal to 1% for each mode. The mode shapes, , are provided by the linear stability theory (LST). β_k and ϖ_k are the corresponding spanwise wavenumber and the radian frequency, respectively. z is the spanwise coordinate in the numerical space. The detailed parameters of the seeded modes are illustrated in Table 1.

(ⅱ) The nonreflecting boundary condition is used on the outflow boundary in the streamwise direction.

(ⅲ) The periodic boundary is used in the circumference direction.

(ⅳ) For the wall boundary, an adiabatic wall temperature together with the assumption is used on the wall, and a second-order, one-side, finite-difference scheme is used to compute the wall pressure.

(ⅴ) The boundary conditions for the upper side of the box are the freestream conditions, i.e., u=ρ=T=1, while v=w=0.

3 Results and discussion

Figure 4 presents the instantaneous flow field with the second mode. The velocity magnitude normalized by the freestream velocity, the dilatation, and its square denoted as θ and θ², respectively, are presented in Figs. 4(a), 4(b), and 4(c) from PIV and 4(d), 4(e), and 4(f) from DNS, respectively. A good agreement is achieved between the PIV and DNS results. Typical 2.8 mm wavelength waves, nearly twice the boundary layer thickness, are significantly presented in the velocity field shown in Fig. 4(a) from PIV and Fig. 4(d) from DNS. Fluids near the wall are periodically accelerated and decelerated along the streamwise direction, indicating a strong compression-expansion. To this end, the divergence of the velocity is calculated in Fig. 4(b) from PIV and Fig. 4(e) from DNS. As shown, a strong periodic compression-expansion process accompanies the evolution of second-mode instability, indicating the acoustic nature of the latter. This process will inevitably bring about dilatational viscous dissipation, denoted as θ², as shown in Fig. 4(c) from PIV and Fig. 4(f) from DNS.

4 Conclusions

In this paper, flow structures of a Mach 6 transitional boundary layers over a 260 mm long flared cone are investigated using PIV. Particle images near the curved wall are initially transformed into surface-fitted orthogonal coordinates and spliced with their 180°-symmetric images to satisfy a no-slip condition at the wall. The interrogated results are then reversely transformed to the physical domain. DNS is also performed to validate the experimental results. The results prove the accuracy of the current PIV near-wall method, which is in good agreement with the DNS results. All the experimental and numerical results exhibit a strong dilatation process within the second-mode instability, which can be identified as the acoustic nature of the latter.