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Disturbance Development in an Obstacle Wake in a Reacting Hypersonic Boundary Layer

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The presented work is a continuation of the investigation of the influence of an isolated roughness on the laminar-turbulent transition for hypersonic boundary-layer flows. The first part of the investigation describing the steady flow around a cuboid obstacle as it was flown in the Hypersonic Boundary-Layer Transition experiment was described by Stemmer et al. [“Hypersonic Boundary-Layer Flow with an Obstacle in Thermochemical Equilibrium and Nonequilibrium,” Journal of Spacecraft and Rockets (in print)]. In the second part, stability investigations reveal odd and even instability modes being unstable in the wake. Direct numerical simulations are used to investigate the disturbance development of high-frequency velocity disturbances. The unsteady, fully three-dimensional direct numerical simulations under consideration of chemical equilibrium and nonequilibrium effects shed light on the role of the chemical models on the disturbance development. The chemical equilibrium case shows less damping of the disturbances in the integration domain compared to the perfect gas case. For the chemical nonequilibrium simulations, the disturbances are initially damped but experience amplification further downstream. Detailed investigations of the disturbance development in the wake flow are presented. The current results underline the necessity of the consideration of chemical nonequilibrium effects in direct numerical simulations of high-temperature boundary-layer transition from laminar to turbulent, even for moderate temperature and dissociation levels.




streak amplitude, m/s


speed of sound, m/s


species concentrations

cph, cgr

phase and group velocity, m/s


frequency, 1/s


altitude, km


integer multiples of the fundamental disturbance frequency


surface-roughness height, mm


integer multiples of the fundamental spanwise wave number


side length of the cuboid obstacle, mm


Mach number


pressure, N/m2


Reynolds number


translational temperature, K


vibrational temperature, K


eigenfunctions of the disturbance temperature from linear stability theory, K


time, s


velocities in x, y, and z directions, m/s


eigenfunctions of the downstream disturbance velocity from linear stability theory, m/s

x, y, z

coordinates in downstream, spanwise, and wall-normal directions, m


normalized spanwise coordinate; y/λy


spatial amplification rate, 1/m


wave number, 1/m


boundary-layer thickness (99% U), m


spanwise (disturbance) wave length, m


boundary-layer momentum thickness, m


dynamic viscosity, kg/(ms)


kinematic viscosity, m2/s


density, kg/m3


shear, 1/s


temporal amplification rate, 1/s

I. Introduction

The presented results are a continuation of a previous paper on the stationary flow around a cuboid mounted on a wedge on top of an experimental rocket described in detail in the work of Stemmer et al. [1]. The steady wake of the cuboid, which is about one-third of the local boundary-layer thickness in height, is subject to instabilities. In direct numerical simulations, disturbances identified to be unstable by the linear stability theory of the steady wake are introduced. In the presence of high-temperature gas effects through chemical (non)equilibrium, the disturbance development in the wake is evaluated.

Cavities and larger-sized isolated roughnesses have been identified as sources of bypass transition early. Reda [2] gave an overview of selected publications in the light of finding a relationship including the roughness height, appropriate boundary-layer, and/or freestream properties to determine a criterion describing immediate transition caused by roughnesses. In reentry scenarios, roughnesses are always present due to the high thermal loads and ablation on the surfaces, as well as manufacturing imperfections (or the so-called “gap fillers” on the space shuttle). Receptivity increases with increased roughness, which can lead to either premature transition (compared to the theoretically smooth case) or immediate (bypass) transition at the location of the roughness. Correlations exist describing this relationship based on experimental data from a phenomenological point of view (in contrast to a stability-theory-based description). Different correlations describing a critical roughness height (Reynolds number) of the Rekk=(ρkukk)/μk type are proposed and evaluated by numerous experiments. The index k denotes the conditions at the roughness height for a flow without roughness (smooth walls), and the value k denotes the roughness height. These curve fits correspond to a transition parameter (Reθ/Me)tr, which is proportional to the inverse of the roughness height nondimensionalized with the momentum thickness of the boundary-layer (k/θ)1. Investigations by Berry et al. [3,4] were used to determine the constant describing the proportionality. These correlation-driven curve fits take into account neither the exact shape of the roughness (cylinder, cube, half-sphere, etc.) nor the resulting shape of the wake of the objects because they investigate for immediate transition at the object itself. Bernardini et al. [5] tried to incorporate the shape of the object into a parameterized transition criterion that allowed the description of immediate (bypass) transition to turbulence at the obstacle. The exact flow deployment downstream of an isolated roughness depends on many factors: mainly, the shape and size of the obstacle determining the longitudinal trailing vortices. For a flight vehicle, this could lead to grooves through increased ablation in the turbulent wedge, which later in the decent is subject to an entirely turbulent flow at lower altitudes.

Choudhari et al. [6] presented an investigation of the (diamond) cuboid roughness element at Ma=3.5 of k=0.55δ, including the far-field wake development with streaks on a flat plate. The wake stability analysis shows even (f=175  kHz) and odd (f=120  kHz) modes being present due to the wall-normal and spanwise shear in the wake.

Wheaton and Schneider [7] experimentally investigated and Bartkowicz [8] numerically investigated a cylindrical roughness in the order of the boundary-layer thickness in a Mach 6 tunnel on a flat plate where a distinct amplified frequency band was identified in the wake region at f=21  kHz with higher harmonics present. They attributed the unstable development in the wake to an absolute instability at the front of the roughness where separated flow was present for these “large” roughnesses. A subsequent publication of the group by Wheaton et al. [9] described the generation of the instabilities at the front of the obstacle and the feeding of the instabilities into the wake. Discrepancies at lower roughness heights were attributed to the noise in the tunnel that was not present in comparable numerical simulations. Chang et al. [10] reported on the strongly accelerated boundary layer on a capsule configuration and simulations under the conditions of [9]. They described the detailed flow around the cylindrical roughness of k/δ1 and the wake at the crew exploration vehicle like body. For the simulations of the Mach 6 wind-tunnel roughness of [9], they noticed an indication of transitional processes at a frequency band around f=46  kHz. Choudhari et al. [11] reinvestigated the Ma=6 case by Wheaton and Schneider [7] and found comparable N factors for the even and odd modes, with the even mode being slightly more strongly amplified. Danehy et al. [12] reported experimental results for Mach=10. For a slightly different setup of the roughness geometry of Choudhari et al. [13], the odd mode was more dominant. The authors noted that the instability development depended heavily on the precise flow conditions and the exact geometry and height of the roughness element. A right-triangle roughness element was shown to be significantly less unstable as compared to a diamond-shaped roughness element with the same height and spanwise width due to the differences in the wake structure.

Redford et al. [14] simulated a tanh-shaped roughness that was half the boundary-layer thickness for Ma=3 and 6 on a flat plate. They attributed transition to the instability of the shear layer forming at the roughness element.

Balakumar [15] reported on receptivity simulations on a high-speed slender cone configuration. Singular roughness elements did not exceed k=1/4δ in height. Slow acoustic and second modes experienced the highest receptivity coefficients. In general, these benign roughness heights did not influence the long-range stability of the boundary layer in the slender cone case significantly. If so, they led to a small stabilization of the boundary-layer flow.

In the work of De Tullio et al. [16], the so-called biglobal stability analysis was applied to the direct numerical simulation (DNS) of a roughness including the wake at a fixed wall temperature at the adiabatic wall-temperature value for a Ma=2.5 flat plate. The cuboid roughnesses were 0.22δ  (Rekk=170) and 0.44δ  (Rekk=791) in height. The small roughness element did not have an effect, whereas the large roughness showed a wake unstable to all frequencies investigated. Even and odd modes were found that lead to laminar-turbulent transition about 50 roughness heights downstream of the roughness location. De Tullio et al. also found that, when even and odd modes showed comparative amplitudes, the disturbance growth was faster than their linear superposition. The results of Choudhari et al. [6] and Kegerise et al. [17] were substantiated.

Park and Park [18] examined the stability of a flat-plate hypersonic boundary layer at Ma=4.5 and 5.92 with a two-dimensional hump (swell) with parabolic stability equations. Upstream of the synchronization point between the fast and slow acoustic waves, the slow acoustic mode was destabilized by the swell. Downstream of the synchronization point, the effect of the swell was stabilizing the slow acoustic mode. Tang et al. [19] and Fong et al. [20] investigated second-mode instabilities downstream of a two-dimensional roughness for a Ma=6 flat plate at ideal gas conditions. The DNS conducted by Marxen et al. [21,22] showed the detailed interaction of primary two-dimensional disturbances with secondary three-dimensional waves with detailed chemistry at Ma=4.8. For a 7 deg cone with a two-dimensional roughness element including high-temperature gas effects by Mortensen [23] and Mortensen and Zhong [24], an increase in amplification rates was noted.

Chaudhry et al. [25] numerically investigated a cylindrical isolated roughness at Ma=6 on a flat plate under quiet and noisy conditions.

Van den Eynde and Sandham [26] undertook a comprehensive study of different roughness shapes in a flat-plate Ma=6 boundary layer concerning their effectiveness on transition due to streak instability in the obstacle wake flow. They identified a strong correlation between the instability mechanism in the wake and the streak amplitude of the wake. This could be used to formulate a more physics-based transition correlation based on the streak amplitude.

The current investigation is looking at the instability of the wake of a roughness in a boundary layer in the presence of high-temperature gas effects in detail. Work on transition correlations as, i.e., Rekk is not detailed enough to understand the origin of the transition process, especially in the cases without bypass transition at the roughness itself. The flight conditions, together with receptivity issues and even wind-tunnel noise in these investigations, play a crucial role in determining the most likely transition scenario for isolated roughnesses. The current paper looks at the influence of chemical reactions and nonequilibrium effects on the development of unsteady disturbances in an otherwise steady wake. Criteria like the “streak amplitude” or the “mean shear” are employed in order to test their correlation with disturbance growth for the different chemical models employed.

First, the experimental setup is briefly presented in Sec. II, which gave the geometrical base for the numerical investigations. The main result in Sec. III consists of the stability investigation for the wake flow as presented in Sec. III.A. Section III.B shows the unsteady behavior of the roughness wake (its steady evolution was described in the work of Stemmer et al. [1]) excited through disturbances introduced at the inflow of the integration domain in the presence of chemical equilibrium and nonequilibrium. The investigation is summed up, and conclusions are presented in Sec. IV.

II. Numerical Procedure and Integration Domain

The governing equations are the complete three-dimensional unsteady compressible Navier–Stokes equations. Through the consideration of the high-temperature gas effects, the thermodynamic coefficients (mainly, but not exclusively, the viscosity μ, the heat capacity cp, and the thermal conductivity λ) are temperature variant and change with the reaction rate of the Park five-species air model. The ideal-gas equation of state is still valid for all chemical cases considered because the pressure is well below 1000 atm and the temperatures present are well above the boiling point for air at 90 K [27]. Additional conservation equations for the chemical species have to be solved, and the constraint that all species concentrations add up to one has to be considered. The solution of the conservation equation for the vibrational energy enables the calculation of chemical and thermal nonequilibrium.

A. Numerical Procedure

The numerical procedure is described in detail in the work of Stemmer et al. [1]. The code called NSMB as used here uses fourth-order-accurate central, finite difference schemes with a third-order-accurate explicit Runge–Kutta time-integration scheme for the direct numerical simulations. The code is capable of finite-rate chemistry calculations, enabling the code to handle chemical equilibrium as well as chemical (and thermal) nonequilibrium (see, i.e., the work of Hoarau et al. [28]).

The wall is modeled with a no-slip condition, and the temperature of the wall is kept constant at T=360  K (see [29]). The upper boundary carries a nonreflecting boundary condition based on Riemann invariants. At the inflow, the flow variables are prescribed because they are calculated from the steady precursor simulations described in [1]. For the unsteady simulations, periodic perturbations are added to the steady solution at the inflow as described in Sec. III.B. At the outflow, supersonic outflow conditions are used. In the spanwise direction, periodic boundary conditions are prescribed. Resolution considerations and grid-refinement studies have been addressed in [1].

B. Geometrical Setup

The simulations covered an experimental configuration described in detail in the work of Berry et al. [29] and Chen and Berry [30]. The roughness elements were mounted on an experimental rocket with a two-dimensional wedge of a half-opening angle of α=6  deg. A cuboid termed “pizza box,” a gap filler, and a cavity had been machined on one side of the wedge at 50.8 cm downstream of the virtual leading edge (see Fig. 1 [31]). The coordinate system was oriented along the upper side of the wedge with x downstream, y spanwise, and z wall normal, as shown in Fig. 1. The other side of the wedge was planar. We chose to simulate an isolated roughness element of the pizza box type with a square base (lh=16.18  mm) and half the side length of the height of the cuboid (h=8.09  mm0.3δ). One side of the cuboid faced the flow. The experiment did not deliver data for comparison with this numerical study because the flight had to be aborted prematurely. Further quantitative data from NASA from preliminary experiments were not released publicly.

Fig. 1
Fig. 1

Two-dimensional cut through the experimental geometry with coordinates; from [31].

C. Integration Domain

The integration domain for the wake simulation consists of 10 equally sized slices in the downstream direction of nine blocks each. The equidistant grid in the spanwise and downstream directions is stretched in the wall-normal direction for a fine resolution in the boundary layer and disturbance damping toward the upper boundary (see Fig. 2; every fourth line is shown). The entire grid consists of more than 36106 grid points in the domain of 0.13  mx0.63  m, 0.041  my0.058  m, and 0  mz0.18  m relative to the right front bottom corner of the obstacle (looking downstream). The obstacle itself is not part of the integration domain any more; only the wake is simulated. Detailed simulations of the flow around the cuboid were conducted [1] to deliver steady flow conditions at x=0.13  m downstream of the cuboid front. These flow conditions for the respective chemical and thermal conditions are used as inflow conditions to the domain described herein. As an example, the downstream velocity u (contour lines 0  m/su2600  m/s, Δu=200  m/s) and the temperature T (1000  KT2800  K) for the ideal-gas case is shown. The frame represents the outline of the cuboid further upstream. Figure 3 clearly shows the mushroom-shaped wake. Further details on the wake and the flow around the cuboid under differing chemical conditions can be found in [1].

Fig. 2
Fig. 2

One slice of the grid for the wake simulations.

Fig. 3
Fig. 3

Steady inflow conditions (velocity contours, temperature as color) for the base flow.

D. Flow Conditions

The freestream Mach number chosen was 8.5 at an altitude of 42.5 km (U=2732.6  m/s), with the last measuring point of the free-flight experiment with the highest velocity during ascent (no measurements at descent had been planned). The postshock Mach number at the location of the roughness was 6.3. The isothermal wall temperature was taken from [29] as TW=360  K based on calculations of the entire configuration. The local flow velocities at the upper front edge of the integration domain (x=0.13  m and z=0.18  m) are shown in Table 1 for the different cases described in this paper. The inflow in the present study is at about 7lh downstream of the cuboid end and extends roughly 30 cuboid base lengths downstream.

III. Results

Before the DNS results of the unsteady disturbance evolution in the obstacle wake are presented in Sec. III.B, biglobal stability analysis results for the steady wake flow will be shown in Sec. III.A.

The DNS results will be shown in three different subcategories for the ideal gas flow (IG), for the chemical equilibrium flow (CEQ), and for the chemical nonequilibrium flow (CNE). The overall temperature level is not high enough for thermal nonequilibrium to play a palpable role in this flow regime.

A. Stability Analysis of the Steady Wake Flow

The possible influence of a wake flow of an obstacle or a cavity (the far-field effect is comparable to a roughness) on the laminar-turbulent transition scenarios depends mainly on the Reynolds number, the size of the obstacle, and its relative size compared to the local boundary-layer thickness, as well as other flow parameters. In the present investigation, we are aiming for the wake becoming unsteady far away from the obstacle (more than five roughness diameters downstream). The possible other wake behavior is a stable wake or, on the other side, a bypass transition immediately at the object itself. The goal of the unsteady flow simulations in the following section is the evaluation of the possible influence of the obstacle-induced vortex pair on the stability of the boundary-layer flow for different levels of chemical modeling.

To investigate the present instabilities of the deformed boundary-layer flow, a biglobal stability analysis of the flow in a plane perpendicular to the main flow direction was conducted by Groskopf et al. [32]. Note that Groskopf et al. used a different nomenclature with y- and z modes. They used the term y mode for instabilities associated with the strong local gradients of u/y, where y was the wall-normal direction in their coordinates (z in this work). Along the line of other works on streak instabilities (e.g., [13]), we want to use the terms even (also termed symmetric or varicose) for the y mode and odd (also antisymmetric or sinuous) for the z mode in [32]. Just upstream of the location where the data of the new inflow have been taken, the flow parameters at x=0.1275  m have been used for the stability analysis. The analysis was made on the base of the ideal gas assumptions Pr=0.72 and γ=1.4. Therefore, the influence of the chemical reactions on the stability properties is only through the change in the main flow profiles (namely, the temperature here) but not through the concentrations. The influence of the chemical reactions at these small dissociation levels on the stability properties is, to a very large extent, due to the change in base flow profiles (mainly T here). The inclusion of the chemical reactions through the consideration of the species conservation equations in the stability equations is not expected to deliver significantly different results.§

Two streamwise wave numbers were chosen that were associated with the first- and second-mode instabilities. For the first-mode instability, the wave number αr,t=1.0 was chosen, which was close to the values known from other flat-plate analyses. For the second-mode instability wave number, αr,t=3.0 was chosen, as this corresponded to λ=(2π/αr,t)2δ. The most unstable eigenvalues were calculated, and their proper eigenfunctions were evaluated. Two different modes could be identified in this manner: first, an even mode, which was associated with strong shear in the wall-normal direction of the vortex-induced flow field. On the other hand, an odd mode was associated with the large base flow gradients in the spanwise direction.

In Fig. 4, the spatial amplification rates are shown for the cases calculated with the stability analysis at x=0.1275  m  (Rx626). For case IG, the highest amplification is for the odd mode, which is consistent with the cold-wall conditions present. For the CEQ case, it is just the opposite, as the even mode shows higher amplification rates. This can be attributed to the fact that the temperatures within the boundary layer are lower. In case of the downstream longevity of the mode, as seen in the work of Groskopf et al. [32], the odd mode seems to be the dominant one of the two. The streaks resulting from the longitudinal vortices exhibit larger gradients u/y in the spanwise direction than u/z in the wall-normal direction.

Fig. 4
Fig. 4

Spatial amplification rates for the even mode (solid line) and the odd mode (dashed line): IG (squares), and CEQ (circles); see the work of Birrer et al. [31].

In Figs. 5 and 6, the eigenfunctions of the downstream velocity u and of the temperature T for the odd and even modes with the strongest amplification for the two cases of IG and CEQ are shown. The values of the stability analysis are normalized to umax=1. The eigenfunctions are shown for the odd mode (top) and even mode (bottom) for αr=3.0. The thin solid lines are isocontours of the base flow: u and T. The thick solid line marks the Mach line Ma=1 within the boundary layer. The phase velocity is in the interval 0.70cph,t0.75.

Fig. 5
Fig. 5

Case IG: eigenfunctions of u (left) and T (right) for the even mode (top) and odd mode (bottom); from [32].

Fig. 6
Fig. 6

As in Fig. 5, but for case CEQ with αr=3.0;0.64cph,t0.68 (from [32]).

The location of the maxima of the eigenfunctions coincides with the location of the maximum base flow gradients.

The instability modes presented in this section are taken as a guideline to model the unsteady inflow disturbances at the inflow of the wake simulation, as presented in the next section.

B. Unsteady Results for the Wake Flow

The spatial development of the disturbance introduced at a potentially unstable frequency of f=100,000  Hz (determined from the stability results) is investigated in this section. The spatial distribution of the disturbances at an instantaneous snapshot is depicted in Fig. 7. The spatial disposition is realized with the help of periodic sin and cos functions, with an exponential decay toward the upper boundary of the integration domain satisfying boundary conditions of the DNS and the stability code. The spanwise and the wall-normal wave numbers are derived from the spanwise extent of the eigenmodes from the stability calculations and slightly adapted for the periodicity of the integration domain in spanwise direction:


Fig. 7
Fig. 7

Unsteady oscillation amplitude of the unsteady downstream velocity component u at the inflow boundary.

These disturbances are then transformed by the stability properties of the flow into the corresponding eigenmodes shown later in the paper. This method to excite disturbance eigenmodes is more effective for high-speed boundary layers than the modeling of a commonly used disturbance strip at the wall, introducing wall-normal velocity disturbances. The spanwise wavelength corresponds to the spatial extent of the eigenmodes shown in Figs. 5 and 6, which is on the order of the width of the obstacle.

For the presentation of the unsteady results, first, the IG case is discussed. Later, the changes observed for the simulations taking into account the chemical reactions are laid out.

1. Case IG

In Fig. 8, the downstream evolution of the velocity u and the temperature T in the range 0.4  m    x    0.6  m is shown. The intensity of the high temperatures in the mushroom-shaped area decreases as expected during its evolution downstream. This results in a decrease in the wall-normal and spanwise gradients of the wake base flow that determine the amplification rates, as was laid out in Sec. III.A. Nevertheless, the amplitudes of the disturbed waves grow on their way downstream past x=0.4  m, as will be shown subsequently.

Fig. 8
Fig. 8

Development of u (top) and T (bottom) within 0.4  mx0.6  m for case IG with inflow disturbances.

For the naming of the disturbance waves, the pair (h,k) is used, where h denotes the integer multiple of the base frequency f=100,000  Hz and k denotes the integer multiple of the fundamental wave number in the spanwise direction (one spanwise wavelength over the entire width of the integration domain). In this nomenclature, the wave shown in Fig. 7 is called (1,6). For the analysis of the disturbances, the unsteady flow is Fourier analyzed in time and in the spanwise direction such that h and k are the harmonics of the fundamental frequency and wave number in the spanwise direction, as all disturbances are periodic in time and spanwise extent. In Fig. 9, mode (1,6) is shown in its downstream evolution after the twofold Fourier analysis in a wall-normal/downstream cut.

Fig. 9
Fig. 9

Amplitude (amp) of the disturbed Fourier mode (1,6) of the downstream velocity u in a wall-normal/downstream cut.

At the inflow boundary at x=0.13  m, the disturbance mode (1,6) is introduced and the overall amplitudes decrease on their way downstream. Downstream of about x=0.35  m, other (nonlinearly generated) modes take over and the wall-normal distribution of the amplitudes (‘amp’ in the figures) changes noticeably, as detailed in Fig. 10 at discrete downstream positions x=0.4, 0.45, 0.5, 0.55, and 0.6 m, respectively. There is still a multitude of modes present that are superimposed. For the downstream disturbance amplitude u(1,6), the maximum amplitude decreases slightly in the interval 0.4  m    x    0.6  m and the location of the maximum moves further away from the wall as the boundary layer grows. The boundary-layer thickness in the respective interval is about δ=0.0387  m. Looking at the wall-normal pressure distribution in Fig. 11, it can be identified as a third-mode disturbance as we recognize four wall-normal pressure phase jumps of more than 90 deg between 0z0.04  m. The inflow disturbances in the area outside the boundary layer can be identified very far downstream, indicated by the high-pressure amplitudes at the boundary-layer edge for x=0.4024  m. The temperature and density amplitudes are rather unaffected and show the amplification in their amplitude development downstream. Downstream of x=0.4  m, the effect of the inflow disturbances on the Fourier amplitudes is vanishing and the typical eigenfunctions of the instability waves clearly show (Fig. 10). The maxima of temperature and density functions reside close to the boundary-layer edge as expected.

Fig. 10
Fig. 10

Wall-normal amplitude profiles of the introduced disturbance mode (1,6) of u, p, T, and ρ.

Fig. 11
Fig. 11

Wall-normal phase profiles of the disturbed Fourier mode (1,6) of pressure p at x=0.4  m.

In Figs. 12 and 13, the wall-normal maxima of the disturbance velocity u (uamp in the figures signifies the fourier amplitude) in their downstream evolution are shown for the three frequency harmonics h=1,2, and 3. With the integration of the relative amplitudes in Eq. (2), the mean amplification rate was determined between x1=0.4  m und x0=0.65  m as


Fig. 12
Fig. 12

Wall-normal maximum of u of the Fourier modes (1,k).

Fig. 13
Fig. 13

Amplitudes of the wall-normal maximum of Fourier modes (2,k) (left) and (3,k) (right) of u. Modes with amplitudes smaller than 103 are not shown.

As the local amplitudes oscillate, the amplification rates given in Table 2 are not the local ones but the average over the downstream range indicated. For the disturbance frequency of 100,000 Hz (h=1), the disturbed mode (1,6) is damped in the range 0.4  m    x    0.6  m (Fig. 12). Mode (1,5) has a notable but decreasing amplitude as well. The modulation of the amplitudes indicates more than one disturbance mode with differing phase speeds present (see the work of Eißler [33]). The most amplified modes are the two-dimensional mode (1,0) with αi=3.81/m and a third mode (1,1) with αi=5.341/m. For the higher-frequency modes (Fig. 13), the largest amplitudes are detectable for two-dimensional modes (2,0) and (3,0). The modes with the same spanwise periodicity as the disturbed mode (h,6) do not play a significant role. The nonlinear interaction of two modes (h1,k1) and (h2,k2) show up as modes (h1±h2,k1±k2) in Fourier space. It might be expected that, e.g., (1,6) and (1,5) would generate a mode (2,11). These modes do not show up in the frequency spectrum with a significant amplitude, which leads to the conclusion that these modes [(1,6) or (1,5)] are modes that have their origin in the disturbance at the inflow and are not unstable themselves in the long run. The frequency higher-harmonics (Fig. 13) show dominant modes with low k in an ascending order (low k have high amplitudes). The two-dimensional mode (2,0) has an amplification rate of αi(2,0)=3.481/m, which is comparable to the amplification rate of mode (1,0). The amplification of mode (2,1) with αi(2,1)=5.581/m is comparable to the amplification rate of mode (1,1). The amplification rates of the respective spanwise modes for the third higher-harmonic frequency (3,0) and (3,1) are even larger because αi(3,0)=5.551/m and αi(3,1)=6.031/m. The largest amplification rates are summed up in Table 2.

The steady modes (0,k) are shown in Fig. 14 (left). The right part of the same figure shows the synthesis of the Fourier modes (0,k) for k0 including mode (0,0) containing the base flow and the steady base flow deviation combined at x=0.55  m (the entire spanwise range would be 0<y¯<1; the absolute width is 0.1 m in dimensional quantities). The influence of the vortex pair in the wake is dominant in Fig. 14 (right) for the steady disturbance modes. On the center line, a low-speed streak has developed at x=0.55  m; very close to the wall, the velocities surpass the mean velocity by up to 346.7  m/s (409.0  m/s for the undisturbed base flow) at y=0.002  m and z=0.0045  m on the outside of the vortex pair. The velocity defect at z=0.01225  m in the center (y=0.008  m) amounts to 832.2  m/s (1252.6  m/s for the undisturbed base flow).

Fig. 14
Fig. 14

Fourier amplitudes of modes (0,k>0) (left) and wall-normal/spanwise distribution of the Fourier synthesis of stationary modes (h=0,k=023) (right).

This can be used to determine the so-called streak amplitude according to

where ubf is the downstream velocity in the base flow without roughness at the location of the maximum/minimum. In the present case, we have determined the streak amplitude from the disturbance quantities because our base flow already contains longitudinal trailing vortices from the obstacle. The disturbance streak amplitude for the IG case amounts to Ast=589.45  m/s (undisturbed: Ast=831.72  m/s). As the shear is a parameter that influences the stability properties directly, we have alternatively calculated the mean shear between the noted maximum and minimum of the streak as

The mean shear in this IG case is τm=88,496.6  1/s.

The spectrum of Fourier modes (0,k) shows dominant uneven modes (0,2k+1). As for mode (1,6), mode (0,6) is negligible in that case. The largest amplitude is for mode (0,1), which is associated with the vortex pair. The nonlinear generation of mode (1,5) from the disturbed mode (1,6) and the stationary mode (0,1) explains the relatively large amplitude of mode (1,5) as well as mode (2,5) [from (2,6)(0,1)(2,5)]; although, in the previous figures, these modes do not show up as amplified modes.

To get a picture of the physical distribution of the modes for the respective frequency, the modes (h,0–23) have been synthesized each for their respective frequency for a wall-normal/spanwise cut at x=0.6  m (Fig. 15). The entire spanwise extent of the integration domain is 0<y¯<1. The spanwise extent of the obstacle is 0.42y¯0.58. For the frequency h=1 (top), noticeable traces of the inflow disturbance are visible because the maxima/minima are spaced in the spanwise distance corresponding to the one of the disturbance mode. The wall-normal maxima are at z=0.02  m at approximately half the boundary-layer height. Closer to the wall, mode (1,1) is identifiable.

Fig. 15
Fig. 15

Wall-normal/spanwise distribution for frequencies h=1 (top), h=2 (center), and h=3 (bottom) at x=0.60  m.

For the first harmonic frequency h=2 (center), the maximum amplitudes are at the very location of the highest shear of the wake vortex pair at z0.0175  m and y¯=0.43;0.57  (y¯=y/λy). This coincides with the results from the instability calculation from the previous section (note that the z axis in Fig. 15 is not to scale). Closer to the wall and to the center line at z0.00875  m and y¯=0.46 and 0.54, two smaller maxima are visible that correspond to the odd mode from the biglobal stability analysis. The odd mode was the one with the largest amplification rate predicted by the stability calculations for case IG, which can be confirmed at hand of the shown DNS results.

For the second harmonic frequency h=3, these maxima show up again in a more pronounced manner. The developing maxima/minima close to the wall are caused by the emerging mode (3,1).

2. Case CEQ

For the case of chemical equilibrium CEQ, only the development of the downstream velocity u is presented. The results that can be drawn from p, T, and ρ are qualitatively similar.

The disturbances introduced at the inflow boundary are the same as in case IG for all cases considered. In Fig. 16, the wall-normal/downstream development of the disturbed Fourier mode (1,6) is shown. The pattern from the inflow disturbance is identifiable further downstream as compared to case IG, which implies a later onset of instability mechanisms or smaller rates of attenuation. Other instability modes take over as far downstream as x=0.5  m. The evolution of the wall-normal profiles of mode (1,6) in x=0.4, 0.45, 0.5, 0.55, and 0.6 m underlines this observation. The maximum amplitude of the disturbed (1,6) mode again decreases between x=0.4  m and x=0.6  m. The mode shape is similar to the IG case. The overall maximum at x=0.6  m in the CEQ case is higher and more pronounced at a wall-normal distance of z=0.02. In the IG case, the amplitude is less than half of the CEQ case and the wall-normal location of the maximum is at z=0.025  m for that case. This can be due to the presence of stronger gradients in velocity and temperature in the steady wake, the further downstream the wake develops relative to the IG case. The CEQ wake rolls up further than the IG wake. The density distributions (Fig. 17, right) are comparable to the IG case because the level of dissociation is rather small altogether.

Fig. 16
Fig. 16

Distribution of u in a wall-normal/downstream cut of Fourier mode (1,6), case CEQ.

Fig. 17
Fig. 17

Wall-normal phase profiles of the disturbed Fourier mode (1,6) of downstream velocities u (left) and ρ (right).

The maximum amplitudes in the wall-normal direction in Fig. 18 (left) along the downstream direction show this behavior as well. The maximum amplitude shows mode (1,6) decreasing over the range 0.4  mx0.6  m. The only modes showing amplification are the two-dimensional mode (1,0) and mode (1,1) with the amplification rates of αi(1,0)=3.961/m and αi(1,1)=4.931/m. A comparable picture to the IG conditions is given by the amplitudes of the higher harmonics in frequencies h=2 and h=3 in Figs. 18 (right) and 19. The amplification rates are similar to the IG case at αi(2,0)=3.801/m for the two-dimensional wave and αi(2,1)=5.871/m. The rates are about 5–10% above those for the IG case for modes (1,0) and (1,1). The downstream amplitude development shows more oscillations compared to the IG case. This effect becomes more pronounced for higher x. In Fig. 20, three wall-normal profiles of mode (1,0) are shown that are taken from downstream locations x=0.562, 0.58, and 0.613 m, where the maximum wall-normal amplitude shows a local maximum, a local minimum, and the next maximum. The eigenfunctions are somewhat different for the maxima and the minimum. In line with the work of Eißler [33], two different eigenmodes that have the same frequency but slightly different downstream wave numbers could be the explanation for that effect. These different eigenmodes end up in the same Fourier mode and superimpose. They can be responsible for the “beating” of the amplitudes as they add up in a more or less favorable way, exposing more of the character of the one mode in the maximum and more of the other mode in the minimum.

Fig. 18
Fig. 18

Wall-normal maximum of Fourier modes (1,k) (left) and (2,k) (right) of u.

Fig. 19
Fig. 19

Wall-normal maximum of Fourier mode (3,k) of the downstream disturbance velocity u.

Fig. 20
Fig. 20

Wall-normal profiles of Fourier mode (1,0) of u.

For mode (3,1), the amplification rate is similar to the IG case. A very strong initial increase in amplitude reflects in an amplification rate of αi(3,1)=12.091/m in the range 0.4  mx0.475  m. If we take the average further downstream at 0.475  mx0.608  m, the amplification rate levels off to αi(3,1)=0.671/m. Mode (3,0) remains at the previously measured level of αi(3,0)=5.901/m. The amplification rates of the described modes are summed up in Table 3.

As in the case IG, the spanwise Fourier modes are synthesized for single frequencies, shown in Fig. 21. The entire spanwise extent of the integration domain is 0<y¯<1. The dimensionless spanwise extent of the obstacle is 0.42y¯0.58. This gives an idea about the distribution of the disturbance in physical space because no single mode is dominating the scenario. For the fundamental frequency h=1, the major amplitudes are found at a wall distance of z=0.02  m. For the first higher harmonic, the (2,1) mode also exhibits larger amplitudes near the wall for 0  mz0.01  m. In contrast to the IG case, the odd mode appears suppressed. Still, the pattern of the introduced disturbance is prominent. Only the higher-harmonic frequency h=3 develops maxima/minima at the locations of high shear in the base flow (z=0.02  m, y¯=0.46 and 0.56).

Fig. 21
Fig. 21

Wall-normal/spanwise distribution of for frequencies h=1 (top), h=2 (center) and h=3 (bottom) at x=0.45  m for CEQ.

The synthesized Fourier amplitudes of u for all steady modes (h=0, k=023) are shown in Fig. 22 (compare Fig. 14 for the IG case). The location of the minimum moves further away from the wall to z=0.0145  m at a velocity deficit of 862.9  m/s (1229.77  m/s for the undisturbed base flow), which is 3.7% larger than for the IG case. The maxima are at the same position as in the IG case but increase to 486.8  m/s (40% higher), which increases the shear at the wall considerably. The value of the maximum for the undisturbed base flow is 397.07  m/s. The streak amplitude for the CEQ case is Ast=674.85  m/s (undisturbed: Ast=813.42  m/s), which is larger by 14.5% than in the IG case. The mean shear in this case is increased to τm=95,589.7  1/s (8%). The relative increase in streak amplitude is twice the increase compared to increase in the mean shear.

Fig. 22
Fig. 22

Wall-normal/spanwise distribution for the frequency h=0 at x=0.55  m.

The effect of the chemical reactions is mainly visible in the temperature development of the main flow. The maximum temperature of the steady base flow is at around T1700  K, which is not high enough to account for a considerable amount of dissociated oxygen. An unsteady amplitude of the temperature disturbances of 10 or a 100 K at this base flow condition would not alter the concentrations because the dissociation levels would not increase noticeably. As an example, the concentration of atomic oxygen in the wake is shown in Fig. 23 for the downstream positions x=0.4, 0.45, 0.5, 0.55, and 0.6 m. In line with the temperature distribution in the wake dominated by the longitudinal vortex, the mushroom-shaped area of dissociated oxygen gets smaller and the level of dissociated gas decreases. A plane parallel to the wall at z=0.007  m shows elevated levels of cO=1.5106 in line with the temperature distribution. The maximum level of dissociated oxygen in this area is cO=6.5106.

Fig. 23
Fig. 23

Concentration of atomic oxygen O in slices perpendicular to the main flow direction at x=0.4, 0.45, 0.5, 0.55, and 0.6 m for case CEQ.

3. Case CNE

The influence of a chemical nonequilibrium flow on the disturbance evolution is presented in the following section.

The downstream/wall-normal distribution of the downstream velocity u (Fig. 24) shows a stronger initial decay of the disturbed modes than in the CEQ case. In disparity to the equilibrium case, downstream of x=0.52  m, the amplitudes rise again, denoting amplification of distinct modes, which could be an indication for nonlinear or transient growth effects.

Fig. 24
Fig. 24

Disturbed Fourier mode (1,6) of u in downstream/wall-normal representation: case CNE.

The wall-normal velocity profiles of the Fourier mode (1,6) for 0.40  mx0.60  m in Fig. 25 show that, in the region close to the wall (z0.02  m), several local maxima develop, which hints at the amplification of other modes than the ones arising in the previously described cases IG and CEQ. The downstream evolution of the wall-normal maximum of the downstream disturbance velocity in Figs. 26 and 27 shows secondary amplification of all higher modes (k>9) at x=0.45  m. The highest amplitudes for x>0.52  m are present in modes (1,0), (1,1), and (1,2). Wall-normal profiles of the higher-frequency harmonics are shown in [34]. Modes (h,0) and (h,1) play a dominant role in the amplitude spectrum other than in the previous cases.

Fig. 25
Fig. 25

Wall-normal profiles of Fourier mode (1,6) of u at representative downstream positions.

Fig. 26
Fig. 26

Wall-normal profiles of Fourier modes (1,k) of u.

Fig. 27
Fig. 27

Wall-normal maximum of Fourier modes (2,k) (left) and (3,k) (right) of the downstream velocity u.

Again, in the cut perpendicular to the main flow direction (yz) at x=0.60  m (Fig. 28, h=1 at the top, h=2 in the middle, h=3 at the bottom), it is noticeable that there is no such local maximum built up as seen in the other cases. There is no distinct present in the spanwise direction. At x=0.60  m, the underlying pattern of the inflow disturbance mode with six maxima and minima in the spanwise direction is still visible. The influence of the cuboid obstacle is seen in the lack of disturbances in the center part of the wake. This is opposite to cases IG and CEQ, where the center wake is the location with the strongest unsteady behavior. It is also conspicuous that the disturbances exhibit much stronger amplitudes close to the wall, which can be attributed to the stronger temperature gradient of the base flow close to the wall for the nonequilibrium case.

Fig. 28
Fig. 28

Wall-normal/spanwise distribution for frequencies h=1, 2, and 3.

For the synthesis of the steady disturbance modes (h=0, k=023) in Fig. 29, the minimum velocity in the center is at 840  m/s (comparable to the IG and CEQ levels by a few percent, cf. Figs. 14 and 22). The maximum moves closer to the wall compared to the CEQ case to z=0.00381  m with a velocity increase of 414  m/s. If one gave a linear estimate of the shear increase at the wall, it was the same as for the CEQ case. The disturbance streak amplitude for the CNE case is Ast=627  m/s (undisturbed: Ast=834.37  m/s), which is larger by 6.3% than in the IG case, which is roughly in the middle between the IG and the CEQ cases. The mean shear in the nonequilibrium case amounts to τm=100,610.4  1/s, which is an increase compared to the IG case of 13.7% (almost twice the increase from IG to CEQ). The streak amplitudes and the mean-shear values for the presented cases are summed up in Table 4.

Fig. 29
Fig. 29

Wall-normal/spanwise distribution for frequency h=0 at x=0.55  m.

To look more closely at this phenomenon, the distinct Fourier modes (1,1), (1,9), and (1,15) in u are shown exemplarily from top to bottom in Fig. 30 in a downstream/wall-normal plane. In this representation, the formation of the local maximum is clearly identifiable at 0.50  mx0.60  m for all the modes.

Fig. 30
Fig. 30

Downstream/wall-normal evolution of the distinct Fourier mode in u for h=1 for three spanwise wave numbers [top (1,1), center (1,9), bottom (1,15)].

In case CNE, the temperature within the boundary layer is larger than for the CEQ case. This results in a higher dissociation rate of oxygen (Fig. 31) as compared to the equilibrium case (Fig. 23) at x=0.4, 0.45, 0.5, 0.55, and 0.6 m. The CNE dissociation levels (the cO levels) are larger than the CEQ levels by a factor of more than three. Through the strong vortical motion of the wake, colder fluid from outside the boundary layer is transported into the wake, which leads to areas with very small dissociated oxygen levels. The present flight case does not deliver high enough temperatures for even higher levels of dissociation. Therefore, the trace of the disturbances in the concentration levels is small and mostly present in the steady local base flow variations. Nevertheless, the disturbance development differs from the CEQ to the CNE cases without large levels of dissociation (the atomic oxygen concentration is in the range of 105). Figure 32 shows the spatial concentration of atomic and molecular oxygen in the flow. The isocontour of a distinct molecular oxygen concentration of c(O2)=0.2346 is colored with the atomic oxygen concentration cO. The inflow plane is colored by the translational temperature. At the inflow plane, the temperature distribution is shown as contours. Both concentrations do not go hand in hand for the nonequilibrium condition. In regions close to the wall, the dissociated atomic oxygen concentration increases. Moving downstream, the level of dissociation decreases, which hints at a decrease in nonequilibrium effects with an increasing distance from the obstacle.

Fig. 31
Fig. 31

Concentration of atomic oxygen O in slices perpendicular to the main flow direction.

Fig. 32
Fig. 32

Concentration isocontour of molecular oxygen O2 at c(O2)=0.2346) in the entire integration domain.

As the disturbance amplitudes, in general, are too small for extended nonlinear interactions, the unsteady flow does not influence the conclusions on the steady-flow wall heat flux, the temperature distribution, or the shape of the wake [1]. Deducing from the larger amplification of higher spanwise wave number modes, one might speculate that the wake would be getting wider. For a long-term evolution of the wake, the integration domain investigated here is not extended enough in the downstream direction because the disturbance amplitudes are too low for a nonlinear developed transition scenario.

IV. Conclusions

A. Conclusions

With the help of a biglobal stability analysis of a local crosscut behind the object, the influence of the longitudinal vortex pair on the wake instabilities was evaluated in the presence of chemical (non)equilibrium. The second mode instability exhibited the highest amplification rates in the presence of the chemical reactions. The inflow disturbances were modeled along the parameters determined from the biglobal stability analysis. The influence of the chemical equilibrium conditions on the spatial deployment of the introduced disturbances in the wake flow of the cuboid obstacle was most noticeable in the higher amplification rates of three-dimensional waves with low obliqueness angles in the linear regime. The so-called odd mode identified in the stability calculation as a possible amplified mode was emerging after a longer runup distance because the amplification rates were moderate.

Considering chemical equilibrium in the DNS of the linear disturbance evolution in the cuboid wake, increased amplification rates of the oblique waves with low spanwise wave numbers (exhibiting maximum amplification in this case) are present. This is consistent with the biglobal stability analysis because the odd mode shows low amplification rates due to smaller local temperature gradients in the flow. This leads to a longer region where the disturbed mode is present before the unstable modes of the wake take over. In all cases, the odd mode takes the lead, which can be attributed to the presence of local disturbance maxima at the locations of the strongest gradient u/y in the base flow. This is comparable to a y mode as known from crossflow instability modes.

For the nonequilibrium case, in contrast to the ideal gas and the equilibrium cases, modes with low and high angles of obliqueness are increasingly amplified in areas far downstream of the obstacle (roughly 33 side lengths lh of the cuboid). In the ideal gas case, the two-dimensional mode k=0 and smallest oblique mode k=1 are dominant. The disturbed mode (1,6) decreases steadily. The equilibrium case shows less decrease of the (1,6) mode, which leads to similar amplitude levels compared to the oblique waves with small angles of obliqueness during later evolution. In the case of nonequilibrium, the high angle of obliqueness waves are amplified more strongly, leading to the highest level of disturbance amplitude in the compared three cases. Modes (1,0), (1,1), and (1,2) possess the largest absolute amplitude in the far field. In the ideal gas and the equilibrium cases, the highest level of disturbances can be found in the wake at the projection of the sides of the obstacle, where the strongest spanwise velocity gradients are present. The nonequilibrium case shows the opposite behavior because the immediate downstream extension of the cuboid is the region with lowest activity.

The classification according to the so-called streak amplitude is not a conclusive indicator for the disturbance development in the presented case for a fixed frequency disturbance. The steady streak amplitude (base flow with no disturbances introduced) and mean shear between the maximum and minimum streaks (as they change in location somewhat) for the steady and disturbance simulations are collected for the three chemical models considered in Table 4.

The picture is inconclusive for both values: the streak amplitude and the mean shear. What is noticeable is that, for both parameters, the values in the disturbed simulations are considerably smaller than for the undisturbed, steady cases. For the streak amplitude (as a reference, the IG case values are taken here), the values drop by about 30%; for the mean shear, the values drop up to 40%. This suggests that both parameters might be an indication for later instability and breakdown to turbulence of an obstacle wake, but they cannot explain the details and the exact potential for disturbance growth or a distance downstream of the obstacle for the disturbance growth to set in.

Although the level of dissociated oxygen is in the parts per million range, a significant change is identified in the disturbance development from the equilibrium to the nonequilibrium cases. For the nonequilibrium case, distinct disturbances grow slowly after an initial attenuation period. As the ideal gas case exhibits higher temperatures but no significant disturbance growth, the overall temperature level is not the origin of the noted behavior. An indication can be the higher level of wall heat transfer with increasing chemical complexity (highest for the nonequilibrium case). The temperature gradient at the cold wall is highest for this case, which might be an indication to the origin of the observed effect of the chemical model on the disturbance development. This is part of the ongoing investigations.

B. Summary

Direct numerical simulations of the unsteady disturbance development in the wake flow of a cuboid obstacle on a wedge including ideal gas conditions and chemical (non)equilibrium in hypersonic flow were conducted under the present conditions for the first time. With the help of a biglobal instability calculation, the unsteady inflow disturbance was fixed to a case where amplification was to be expected. The disturbance development in the far field of the obstacle wake showed an indication of less attenuation in the chemical equilibrium case as compared to the ideal gas case. The low obliqueness waves were especially slightly amplified in the long run. A late disturbance growth of oblique waves with low and higher angles of obliqueness was present in the chemical nonequilibrium case. The pure level of the so-called streak amplitude was not a conclusive measure to explain the different behavior in the far field of the obstacle including chemical reactions and nonequilibrium. The presented case underlined the necessity to investigate the instabilities in the wake of an isolated roughness in a high-enthalpy boundary layer under consideration of chemical nonequilibrium, even at moderate temperatures and small dissociation levels.

R. KimmelAssociate Editor

§ Personal communication with G. Candler, University of Minnesota, June 2005.


Sponsorship and financial support of this work was generously granted through the International Graduate School of Science and Engineering at the Technische Universität München through grant number 2.01. The cooperation with Markus Kloker and Gordon Groskopf at the Institute for Aero- and Gasdynamics of the Universität Stuttgart for the stability results is greatly appreciated. Computing time supporting the presented work has been provided by the Leibniz-Rechenzentrum Munich, Germany and the Höchstleistungsrechenzentrum Stuttgart, Germany on their high-performance computers. The discussions with the people from NASA involved in the Hypersonic Boundary Layer Transition experiment (namely, Scott Berry from NASA Langley) are greatly acknowledged. We would also like to express our thanks to the reviewers who spent their valuable time pointing out the right issues to help in improving this paper.


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Table 1 Flow conditions at inflow at x=0.13  m downstream of the cuboid front

Caseρ, kg/m3u, m/sT, Kp, N/m2Ma

Table 2 Amplification rates of the modes with strongest amplification of u in the range of 0.4  mx0.6  m for the spanwise wave number multiples k=0 (two-dimensional) and k=1, in units of 1 per meter


Table 3 Amplification rates of the modes with strongest amplification in the range of 0.4  mx0.6  m for (h,0) and (h,1): case CEQ, in units of 1 per meter

Range[0.4 m; 0.47 m][0.47 m; 0.6 m]

Table 4 Streak amplitude and mean shear in the wake at x=0.13  ma

CaseAst steady, m/sAst disturbed, m/sτm steady, 1/sτm disturbed, 1/s
CEQ813.42 (2.2%)674.85 (+14.5%)149,736.00 (+0.2%)95,589.7 (+8.0%)
CNE834.37 (+0.3%)627.03 (+6.3%)142,506.37 (4.7%)100,610.4 (+13.7%)

aPercentages are relative to the respective IG case value.