Numerical Investigation of Roughness Effects on Transition on Spherical Capsules

A. Optimal Transient-Growth Theory

The optimal transient-growth analysis is performed using the framework of linear parabolized stability equations (PSE) as elucidated in the literature [ 19–22]. The method is outlined here for the sake of completeness.

4. Cross Comparison of the Optimal Transient-Growth Codes

The nonmodal disturbance growth results presented in this paper have been computed with two different codes. The optimal-growth framework developed by NASA is used for the computations for the TAMU ACE capsule configuration and has been extensively verified [ 21]. On the other hand, a newly developed optimal transient-growth code by DLR is employed for characterizing the nonmodal growth properties of the boundary layer on the HLB capsule. A more detailed overview of the two different capsule configurations is given in Sec.  III.

The boundary-layer flow over a hemisphere at $Ma=7.32$ is considered to cross-verify the two different transient-growth implementations used in this work. Details on the basic flow computations by NASA were reported by Li et al. [ 32], and the nonmodal disturbance growth characteristics of the boundary layer in downstream direction are given in Ref. [ 22]. Figure  1 depicts the streamwise evolution of basic flow variables at the edge of the boundary layer along the angular coordinate $\varphi$ ( $\varphi =\xi /R_s$ with $R_s$ being the radius of the hemisphere). The boundary-layer edge is defined as the wall-normal position where the total enthalpy reaches 99.5% of the freestream value ( $h_t/h_{t,\infty }=0.995$). The basic state computed by DLR uses the numerical framework described in Theiss et al. [ 8]. Figure  2 shows the optimal outlet energy gain based on the total energy of the disturbance, $G_E$, at a fixed output location, ${\varphi }_1=32.2°$, and varying inflow positions, ${\varphi }_0$, as a function of the azimuthal wavenumber, $m_{\zeta }$, for the basic state computed by DLR. An excellent agreement of the predicted gain from both codes is observed.

B. Direct Numerical Simulations

The numerical method used by RWTH solves the compressible Navier–Stokes equations in space and time [ 33]. The computational domain is discretized by an unstructured hierarchical Cartesian mesh whose cells are arranged in an octree structure. The domain decomposition for parallel computations is based on the Hilbert space-filling curve and subtree workloads to distribute subtrees of the hierarchical octree of equal loads to processors [ 34]. The governing equations are integrated using a finite-volume method [ 35]. The boundaries of the computational domain are embedded in the Cartesian mesh and modeled employing cut cells [ 33]. Small cut cells are treated using an interpolation and flux-redistribution scheme [ 35].

For the spatial discretization, an advection upstream splitting method (AUSM) is used. The advection Mach number on the cell surface is the mean of the extrapolated Mach numbers from the adjacent cells ${Ma}_{1/2}=0.5({Ma}_L+{Ma}_R)$. The same formulation holds for the pressure on the cell surface. The cell-center gradients are computed using a second-order accurate least-squares reconstruction scheme [ 33]. Shock capturing is achieved by adding additional numerical dissipation at the shock position. The temporal integration is based on a five-stage second-order accurate Runge-Kutta scheme. For supersonic flows, the code has been employed and validated for the flow past a cone and around a blunt stagnation point probe [ 36, 37].

The DNS by the Technical University of Munich (TUM) are performed using the Navier–Stokes Multi Block solver (NSMB). NSMB is an MPI-parallelized, finite-volume code for structured grids with a wide variety of numerical schemes, and it has been extensively tested in studies of hypersonic flows [ 38, 39]. The spatial discretization is based on a fourth-order, central difference scheme, whereas a fourth-order Runge–Kutta method is used for time integration. Artificial numerical dissipation is added to capture the shock and to suppress spurious oscillations.

A. TAMU Capsule

The first configuration for which transient-growth results are presented corresponds to a blunt, spherical-section forebody at 28° angle of incidence with respect to the free stream. The forebody configuration models the Orion CEV capsule geometry [ 40]. The face diameter is $D=2R=0.0762\mathrm{m}$ and the remaining dimensions are scaled according to Hollis et al. [ 40], resulting in a sphere of radius $R_s=0.09144\mathrm{m}$. The flow conditions at Mach 6 match those of a wind-tunnel experiment in the Actively Controlled Expansion (ACE) tunnel at the National Aerothermochemistry Laboratory (NAL) of TAMU [ 13, 14]. Paredes et al. [ 12] performed transient growth calculations for four freestream unit Reynolds numbers, namely, $Re/l=3.4\times {10}^6/\mathrm{m}$, $4.4\times {10}^6/\mathrm{m}$, $5.4\times {10}^6/\mathrm{m}$, and $6.4\times {10}^6/\mathrm{m}$. The freestream temperature was set to ${\overline{T}}_{\infty }=54.69\mathrm{K}$ and the surface temperature was equal to ${\overline{T}}_w=391.0\mathrm{K}$. To investigate the effects of surface temperature on the transient-growth characteristics, computations were also performed for additional, cooler surface temperatures corresponding to ${\overline{T}}_w=300.0$, 195.5, and 130.33 K, respectively, with the unit Reynolds number held fixed at $Re/l=4.4\times {10}^6/\mathrm{m}$.

The basic state, laminar boundary-layer flow over the forebody was computed by using a second-order accurate algorithm as implemented in the finite-volume compressible Navier-Stokes flow solver VULCAN-CFD (see Ref. [ 41] and http://vulcan-cfd.larc.nasa.gov for further information about the solver). Further details about the laminar basic flow computations are given in Ref. [ 12].

B. HLB Capsule

The second capsule studied in this work corresponds to an Apollo-shaped capsule with a spherical-section forebody ( $D=0.17\mathrm{m}$ and $R_s=0.204\mathrm{m}$) at an angle of attack of $\mathrm{AoA}=24°$. Laminar basic flow computations are performed for Mach 5.9 freestream conditions that match the experiments of Ali et al. [ 15] in the HLB. Overall, four different unit Reynolds numbers have been computed, namely, $Re/l=10\times {10}^6/\mathrm{m}$, $12.5\times {10}^6/\mathrm{m}$, $16\times {10}^6/\mathrm{m}$, and $18\times {10}^6/\mathrm{m}$, with the freestream temperature set to ${\overline{T}}_{\infty }=59.03\mathrm{K}$ and prescribed surface temperature of ${\overline{T}}_w=295\mathrm{K}$. To also assess the impact of surface temperature on the nonmodal disturbance growth characteristics, additional simulations have been performed at fixed freestream conditions for $Re/l=10\times {10}^6/\mathrm{m}$ and modified surface temperatures with ${\overline{T}}_w=170$, 245, and 395 K, respectively.

The laminar basic flow was computed with the second-order accurate, three-dimensional, finite-volume, compressible Navier–Stokes flow solver FLOWer [ 42] on a block-structured grid. More details about the numerical settings and the employed grid are given in Ref. [ 8].

C. Comparison of Boundary-Layer Edge Data for the Two Capsules

Figure  3 shows a three-dimensional view of the HLB capsule forebody. Mach number isocontours in the symmetry plane and the Reynolds number based on momentum thickness, ${Re}_{\theta }$, with boundary-layer edge streamlines on the capsule forebody are plotted for $Re/l=10\times {10}^6/\mathrm{m}$ and ${\overline{T}}_w=295\mathrm{K}$. The thick dashed lines indicate the sonic lines. Only one half of the model was used in the basic flow computations, exploiting the azimuthal symmetry of the flow field. Because of the strong bow shock, the boundary-layer edge Mach number falls within the subsonic to transonic range and the flow continuously accelerates from the stagnation point at $\xi =0$ toward the capsule shoulder. In this work, we focus on the leeward symmetry region above the stagnation point, where transition has been observed in experiments [ 8, 14]. The HLB capsule has been investigated at higher unit Reynolds numbers, and its diameter is about twice the size of the Orion CEV model. The angle of attack differs also, but due to the spherical forebody, the boundary-layer edge values normalized with the respective freestream values are very similar for the two flow configurations as depicted in Fig.  4. Boundary-layer edge quantities along the symmetry plane for the HLB capsule at $Re/l=10\times {10}^6/\mathrm{m}$ and ${\overline{T}}_w=395\mathrm{K}$ are compared with those for the TAMU capsule at $Re/l=4.4\times {10}^6/\mathrm{m}$ and ${\overline{T}}_w=391\mathrm{K}$. In accordance with Sec.  II.A.4, the boundary-layer edge is determined from the total enthalpy criterion ( $h_t/h_{t,\infty }=0.995$). For both capsule geometries, the mass flux, ${\overline{\rho }}_e{\overline{u}}_e$, increases with growing distance from the stagnation point and reaches its maximum at the sonic point, ${Ma}_e=1$, in agreement with the inviscid flow theory.

D. Hemisphere Approximation

To ease the computational effort in the case of unsteady DNS, a reduction of the domain size is performed in the simulations of TUM as shown in Fig.  5. First, the flow over a hemisphere is considered. The flow over the hemisphere well represents the flow over the HLB capsule with angle of attack with minor restrictions outside of the area of interest. Second, results for the laminar steady flow on the whole hemispheric forebody (full domain) are used to generate inflow profiles for a restricted computational domain. The grid resolution of the restricted domain can be increased to match the resolution requirements imposed by the presence of the rough wall. Simulations for the smooth configuration are conducted on the entire hemisphere (full domain), whereas simulations for roughness investigations are conducted on a restricted domain. The box in Fig.  5a shows the position of the restricted domain. The roughness position is indicated by the red square in Fig.  5b.

The steady base flow for the entire hemisphere with smooth surface is computed on an axisymmetric two-dimensional grid. The grid consists of about 76,000 points clustered around the shock location and inside the boundary layer. To provide similar outflow boundaries as in the case of the reentry capsule, the hemisphere ends with a shoulder resembling the one of the HLB capsule.

For both the 3D capsule and the axisymmetric 2D hemisphere, boundary-layer streamwise velocity and temperature profiles at different positions are shown in Fig.  6 for a unit Reynolds number of $Re/l=18\times {10}^6/\mathrm{m}$. The profiles on the 3D capsule geometry are extracted along the symmetry plane. The origin of the streamwise coordinates $s$ and $\widehat{s}$ is set on the stagnation point of the respective configuration. A clear match of the profiles is observed for the two configurations. In particular, the equivalence of the two flows is obtained by comparing velocity and temperature profiles on the capsule at a given position $s$ with the ones on the hemisphere at $\widehat{s}=s+\mathrm{\Delta }s$, with $s$ and $\widehat{s}$ being the streamwise coordinates on the two geometries with origin on the respective stagnation points. On the capsule, the distance between the stagnation point and the rotation axis is $s=60.5\mathrm{mm}$. In the vicinity of this position, the flow corresponds to the one on the hemisphere at $\widehat{s}=77.4\mathrm{mm}$. Details on the restricted domains and further comparisons between the capsule and the hemisphere boundary layer in presence of roughness are discussed in Sec.  V.B.1.

A. Effects of Unit Reynolds Number and Energy Norm

Paredes et al. [ 12] pointed out that, in order to apply the optimal transient-growth predictions toward transition correlations for nonsimilar boundary-layer flows such as the HLB and TAMU capsules, both the initial and final locations of the transient-growth interval must be varied in addition to the azimuthal wavenumber of the disturbance. First, we address the impact of energy norm (total energy vs kinetic energy only) on the gain based on outlet energy [Eq. ( 3)] at $Re/l=10\times {10}^6/\mathrm{m}$ and ${\overline{T}}_w=295\mathrm{K}$. The maximum gain within all possible optimization intervals $[{\varphi }_0,{\varphi }_1]$ ( ${\max }_{[{\varphi }_0,{\varphi }_1]}G$) at the corresponding optimal azimuthal wavenumber, $m_{\zeta ,\mathrm{opt}}$, is plotted for the total energy norm in Fig.  7a and for the kinetic energy norm in Fig.  7b with $J=G^{\mathrm{out}}$, respectively. The black dot indicates the location of the optimal interval where the highest gain occurs, ${[{\varphi }_0,{\varphi }_1]}_{\mathrm{opt}}$. The black line denotes the value of ${\varphi }_1$ corresponding to maximum $G_E$ and $G_K$ for a given ${\varphi }_0$. The region included in the figure is limited by the line of zero length optimization interval ${\varphi }_1={\varphi }_0$ with $G=1$ on the diagonal, and a line on the left that delimits the region of the $[{\varphi }_0,{\varphi }_1]$ space that is omitted because the initial disturbance profiles at ${\varphi }_0$ peak above the boundary-layer edge and do not exhibit sufficient decay in wall-normal direction (especially the wall-normal velocity component), which in turn prevents the adjoint-based optimization algorithm to converge toward a satisfactory result. However, the excluded portion of the plot is not considered to be important for the present analysis because perturbations with an extended wall-normal support are unlikely to be excited by surface roughness. The maximum disturbance energy gain in the case of $J_E^{\mathrm{out}}$ occurs close to the stagnation point, which is in line with the observations for a hemisphere in hypersonic flow [ 22, 27] and the TAMU capsule with ${\overline{T}}_w/{\overline{T}}_e<1$ [ 12]. When the norm for the optimization is based on the kinetic energy alone ( $J_K^{\mathrm{out}}$), the location of the maximum gain shifts further downstream toward the vicinity of the sonic point as depicted in Fig.  7b ( ${\varphi }_{{Ma}_e=1}\approx 35°$; see also Fig.  4), which again is in close agreement with previous observations for blunt body configurations [ 12, 22].

From here on, the overall nonmodal growth characteristics of the flow are presented in terms of optimal combination of azimuthal wavenumber, $m_{\zeta ,\mathrm{opt}}$, and optimal optimization interval length, ${[{\varphi }_0,{\varphi }_1]}_{\mathrm{opt}}$, that lead to the maximum value of the energy gain for a specific initial location. In the following, the impact of unit Reynolds number on the optimal transient-growth characteristics is discussed with respect to objective functions based on outlet energy and mean energy, respectively. The former objective function maximizes the disturbance energy at a prescribed outlet location ${\varphi }_1$ [see Eq. ( 3)], whereas $J=G^{\mathrm{mean}}$ maximizes the integral energy in the interval $[{\varphi }_0,{\varphi }_1]$ [see Eq. ( 4)], which can lead to a higher possible overshoot in the disturbance energy evolution in comparison to $J=G^{\mathrm{out}}$.

Figure  8 depicts the evolution of the maximum disturbance energy gain along the angular coordinate for the four possible optimization options ( $J_E^{\mathrm{out}}$, $J_E^{\mathrm{mean}}$, $J_K^{\mathrm{out}}$, and $J_K^{\mathrm{mean}}$) at different unit Reynolds numbers and ${\overline{T}}_w=295\mathrm{K}$. Note that the region below ${\varphi }_0<6°$ for $G_E$ and ${\varphi }_0<11°$ for $G_K$ is omitted, respectively. In those areas, the length of the optimization interval is limited by the boundaries of the parameter space depicted in Fig.  7. In the case of mean energy gain as the objective function for optimization, the length of the optimal optimization interval is much longer than when the objective function corresponds to the outlet energy gain. Therefore, a meaningful comparison of both objective functions with regard to the highest possible disturbance energy gain is not feasible below the mentioned regions. For all cases shown in Fig.  8, maximizing the outlet energy, $J^{\mathrm{out}}$, leads to the highest possible disturbance energy gain within ${[{\varphi }_0,{\varphi }_1]}_{\mathrm{opt}}$ at $m_{\zeta ,\mathrm{opt}}$. On that account, all of the results presented below to define the upper bound of optimal transient growth for the HLB capsule will pertain only to the objective function based on outlet energy gain ( $J=G^{\mathrm{out}}$). For all unit Reynolds numbers considered here, the total energy gain of the perturbations reduces with increasing distance from the stagnation point, whereas the kinetic energy gain grows toward the shoulder of the capsule, which in turn implies an increasing share of the overall energy. The sudden decay in energy gain at ${\varphi }_0\approx 37°$ for both energy norms, $G_E$ and $G_K$, is attributed to the shortened optimization interval length at the end of the simulation domain (see Fig.  7).

From the optimal transient-growth analysis for boundary-layer flows over flat plates, the disturbance energy gain at high Reynolds numbers is known to scale with the body-length Reynolds number ${Re}_L$ [ 27, 43]. Figure  9 indicates the nearly linear scaling of the optimal total disturbance energy gain ( $G_E$) with the unit Reynolds number ( ${Re}_R=R\cdot Re/l$). The results shown in Fig.  9 are for $J_E^{\mathrm{out}}$ and ${\overline{T}}_w=295\mathrm{K}$. Because the dimension of the body is kept constant when the unit Reynolds number is varied, the observed small deviations from the linear trend are attributed to the differences in the ratio of boundary-layer thickness to the radius of surface curvature. The linear-like unit Reynolds number dependency of the optimal disturbance energy gain is also reported for the TAMU capsule [ 12].

The transient-growth amplification with regard to the logarithmic amplification ratio, that is, $N$-factor, for ${\overline{T}}_w=295\mathrm{K}$ is shown in Fig.  10 in terms of $N$-factor envelope curves and ${\max }_{{[{\varphi }_0,{\varphi }_1]}_{\mathrm{opt}}}(N)$ optimized for $J^{\mathrm{out}}$ with optimal outlet energy norms based on total and kinetic energy, respectively. The $N$-factors based on the total energy norm $N_E$ and kinetic energy norm $N_K$ are defined as

The vertical dashed lines in Fig.  10 indicate the transition locations for the experiments of Ali et al. [ 15]. The $N$-factors at the observed transition location based on the norm for total disturbance energy and kinetic energy are $N_E=2.54$ and $N_K=2.45$ at $Re/l=16\times {10}^6/\mathrm{m}$ and $N_E=2.61$ and $N_K=2.48$ at $Re/l=18\times {10}^6/\mathrm{m}$, respectively.

The optimal transient-growth results in Figs.  8– 10 have been presented at the optimal parameters of spanwise wavelength and optimization interval. Figure  11a depicts the optimal spanwise wavelength for $J_E^{\mathrm{out}}$ in terms of boundary-layer thickness along the initial optimization locations for the HLB capsule with ${\overline{T}}_w=295\mathrm{K}$ and for four different Reynolds numbers. In addition, the optimal parameters for the TAMU capsule with ${\overline{T}}_w=391\mathrm{K}$ are also plotted at the unit Reynolds numbers from Ref. [ 12]. For both capsule geometries, the optimal disturbance wavelength displays a good scalability with the boundary-layer thickness. The optimal wavelength with respect to the boundary-layer thickness varies in the range of ${(\lambda /\delta )}_{\mathrm{opt}}\approx [1.6,2.7]$ for the HLB capsule and ${(\lambda /\delta )}_{\mathrm{opt}}\approx [2.2,3.0]$ for the TAMU capsule (in the region without domain boundary effects) and is not too different from the findings of Reshotko and Tumin [ 11] with ${(\lambda /\delta )}_{\mathrm{opt}}\approx [3,3.5]$ for the flat plate and ${(\lambda /\delta )}_{\mathrm{opt}}\approx 3.2$ for stagnation point flows. Although the predicted optimal wavelengths are similar for both configurations, the small difference may have been caused by the different wall temperatures used in the two studies ( ${\overline{T}}_{w,\mathrm{HLB}}=295\mathrm{K}$, ${\overline{T}}_{w,\mathrm{TAMU}}=391\mathrm{K}$) and the resulting ratios of wall temperature to boundary-layer edge temperature ( ${({\overline{T}}_w/{\overline{T}}_e)}_{\mathrm{HLB}}<1$ and ${({\overline{T}}_w/{\overline{T}}_e)}_{\mathrm{TAMU}}\approx 1$). The impact of the ${\overline{T}}_w/{\overline{T}}_e$ ratio on the optimal parameters for nonmodal disturbance growth will be addressed in the next subsection. Figure  11b shows the length of the optimal transient-growth interval as a function of the initial location for the HLB and TAMU capsules at the respective unit Reynolds numbers. Even though the dimension of the TAMU capsule is only about one half the size of the HLB capsule, the optimal length of the transient-growth interval is nearly the same for both geometries ( ${({\xi }_1-{\xi }_0)}_{\mathrm{opt}}\approx 1.0\mathrm{cm}$) and decreases slightly with the unit Reynolds number. The relatively short optimal optimization length is consistent with the findings of Theiss et al. [ 17, 18] for the laminar wake flow development behind an isolated roughness element on the forebody of the HLB capsule. The authors have shown that, due to the strongly favorable pressure gradient [ 44], the laminar wake flow experiences growth and decay of the streak amplitude (and also modal disturbance growth) only within a few roughness diameters downstream of the element. Although not shown here, choosing $J_E^{\mathrm{out}}$ as the objective function, the boundary layer on both capsule forebodies undergoes optimal nonmodal disturbance growth within 30–40 boundary-layer thicknesses depending on the angular coordinate, which is about 15 times shorter in comparison to the findings of Reshotko and Tumin [ 11] for flat plate flows.

B. Effect of Wall Temperature

The effect of the wall temperature on the optimal disturbance growth based on the total and kinetic energy norm is shown in Fig.  12 for the TAMU capsule with $Re/l=4.4\times {10}^6/\mathrm{m}$ optimized for the outlet energy gain. The corresponding results for the HLB capsule with $Re/l=10\times {10}^6/\mathrm{m}$ are plotted in Fig.  13. In accordance with previous findings in the literature [ 9, 11, 20, 27, 45–47], the disturbance energy gain increases with wall cooling, whereas the effect is more pronounced in the case of the total energy norm, especially in the vicinity of the stagnation point. The disturbance energy gain based on total and kinetic energy norms is higher for the HLB capsule due to the larger body-length Reynolds number by a factor of about five. The share of the kinetic energy on the total energy of the disturbance increases with ${\overline{T}}_w/{\overline{T}}_e$ and for ${\overline{T}}_w/{\overline{T}}_e\approx 1$ (HLB capsule: ${\overline{T}}_w=395\mathrm{K}$; TAMU capsule: ${\overline{T}}_w=391\mathrm{K}$; see also Fig.  4) the total energy mainly consists of kinetic energy, that is, at ${\varphi }_0=35.5°$, $G_E=70.6$ and $G_K=69.7$ for the HLB capsule, and $G_E=10.3$ and $G_K=9.9$ for the TAMU capsule, respectively.

The effect of wall temperature on the optimal spanwise wavelength and the optimal optimization interval for both capsules is shown in Figs.  14a and 14b, respectively. The objective function is $J_E^{\mathrm{out}}$. The optimal azimuthal wavelength scaled by the boundary-layer thicknesses decreases slightly with wall cooling for both capsule configurations. The results for the TAMU capsule at $Re/l=4.4\times {10}^6/\mathrm{m}$ fall within the range of the HLB capsule data for $Re/l=10\times {10}^6/\mathrm{m}$ when ${({\overline{T}}_w/{\overline{T}}_e)}_{\mathrm{TAMU}}<1$. On the other hand, the optimal optimization length increases slightly with wall cooling ( ${({\xi }_1-{\xi }_0)}_{\mathrm{opt}}\approx 1.1\mathrm{cm}$) along with a higher deviation from the mean value.

C. Revision of Transient-Growth-Based Transition Correlation

Recently, Paredes et al. [ 12] revisited the distributed roughness-induced transition correlation of Reshotko and Tumin [ 11], which is the only physics-based model that tackles the blunt-body paradox. The RT-correlation is defined as

Assuming a power-law variation of the scaled optimum transient energy gain with respect to the surface-to-edge temperature ratio,

Reshotko and Tumin [ 9, 11] in the vicinity of ${\overline{T}}_w/{\overline{T}}_e\approx 0.5$ obtained a value for the power-law exponent of $c_T=-0.77$, which finally yields Eq. ( 11). For their analysis they used optimal transient-growth computations based on local, parallel theory and self-similar boundary-layer flow without curvature effects. Furthermore, the initial optimization position and the spanwise wavenumber also remained unchanged. Paredes et al. [ 12] applied an advanced framework to improve the shortcomings of the optimal transient-growth computations by Reshotko and Tumin [ 9, 11], namely, nonlocal transient growth computations including curvature effects for full Navier–Stokes basic state solutions of the TAMU capsule at varying wall temperatures ${({\overline{T}}_w/{\overline{T}}_e)}_{\mathrm{TAMU}}<1$. The initial ( ${\xi }_{\mathrm{in}}={\xi }_0$) and final locations ( ${\xi }_{\mathrm{tr}}={\xi }_1$), as well as the spanwise wavenumber, were also optimized. Based on the improved framework, Paredes et al. [ 12] revised the original RT-correlation and also assumed a power-law variation for the optimum transient energy gain with respect to the surface-to-edge temperature ratio:

In their analysis, they computed the $c_T$ value for several possible transition onset locations, ${\varphi }_{\mathrm{tr}}$, based on the optimal parameter combinations ( ${\lambda }_{\mathrm{opt}}$, $m_{\zeta ,\mathrm{opt}}$, ${({\varphi }_{tr}-{\varphi }_0)}_{\mathrm{opt}}$) for each of the different wall temperatures involved ( ${\overline{T}}_w/{\overline{T}}_e<1$) and for maximizing the total energy gain ( $J_E^{\mathrm{out}}$). Moreover, three assumed variations in spanwise disturbance wavelength were also considered. As a result, the set of exponents was nearly insensitive to the assumed $\lambda$ variation and all $c_T$ values were averaged, resulting in a mean value of $c_T\approx -0.81$, which is remarkably close to the value computed by Reshotko and Tumin [ 9, 11] based on parallel flow transient-growth calculations. For that reason, the following revised RT-correlation of Paredes et al. [ 12] only slightly deviates from the originally proposed correlation [Eq. ( 11)]

The purpose of this section is not to provide an additional transient-growth-based transition correlation, but rather to check if the presented nonmodal growth data for the HLB capsule will result in a similar power-law exponent, $c_T$, as derived based on the TAMU capsule data. The results shown in Fig.  13a are used to estimate the best-fit exponent through the relation given in Eq. ( 14). Note that results were considered only when ${\overline{T}}_w/{\overline{T}}_e<1$, in particular, ${\overline{T}}_w=170\mathrm{K}$, 245 K, and 295 K. Figure  15 depicts the variation of the $c_T$ value at selected transition onset locations for the HLB capsule with $Re/l=10\times {10}^6/\mathrm{m}$. In addition, the results for the TAMU capsule at $Re/l=4.4\times {10}^6/\mathrm{m}$ with the spanwise disturbance wavelength variation based on axisymmetric flow (the same assumption as used in this work) are also shown. The angular coordinate for the HLB capsule data is shifted by $\mathrm{\Delta }\varphi \approx 1.3°$ to match the boundary-layer edge Mach number conditions on the TAMU capsule (see Fig.  4). The averaged power-law exponent for the HLB capsule data is $c_{T,\mathrm{HLB}}=-0.813$ and therefore in very good agreement with the equivalent TAMU capsule value of $c_{T,\mathrm{TAMU}}=-0.809$.

A. HLB Capsule Roughness Simulations by RWTH

The roughness patches are located in the center of the spherical forebody of the generic Apollo-like “HLB capsule” in Fig.  3. The origin of the spherical coordinate system is set in the center of the roughness patch, with $\xi$, $\zeta$, and $\eta$ being the streamwise, spanwise, and wall-normal direction, respectively. At the given location, the Mach number at the edge of the boundary layer is ${Ma}_e=0.5$. For the current Reynolds number $Re/l=6.25\times {10}^6/\mathrm{m}$, the boundary-layer thickness, displacement thickness, and momentum loss thickness are $\delta =492\mu \mathrm{m}$, ${\delta }^{*}=89.2\mu \mathrm{m}$, and $\theta =74.3\mu \mathrm{m}$, respectively. In this section, the boundary-layer edge is defined as the wall-normal distance where the flow has 99% of the total enthalpy of the freestream. The Reynolds number based on the displacement thickness is ${Re}_{{\delta }^{*}}=109.65$. The roughness patch elements protrude $20\mu \mathrm{m}$ into the boundary layer, which is 4% of the boundary-layer thickness and 22% of the displacement thickness. The corresponding roughness Reynolds numbers are ${Re}_{\mathrm{kk}}=5.82$ and ${Re}_k=6.02$, respectively.

The geometry of the aforementioned finite roughness patches is illustrated in Fig.  16, where the flow direction at the boundary-layer edge is from left to right. Each of the patches consists of a certain number of identical single roughness elements having a square base of $100\mu \mathrm{m}\times 100\mu \mathrm{m}$ and a height of $20\mu \mathrm{m}$. Two patterns of elements are considered. In Fig.  16a, the “aligned” or “checkerboard” configuration with $5\times 5$ elements is depicted. Based on the flow direction, neighboring elements in the spanwise direction form rows, whereas columns are found in the streamwise direction. This configuration possesses infinite channels between the columns of elements. The spacing between the elements in both streamwise and spanwise direction is $L=200\mu \mathrm{m}$. The “staggered” configuration is shown in Fig.  16b. The second and fourth rows of elements are reduced by one element and misaligned in the spanwise direction by $L/2$. The projected area in the streamwise direction is gapless. The base area of both patches is $0.9\mathrm{mm}\times 0.9\mathrm{mm}$.

To resolve the micron-sized roughness patch, the unstructured Cartesian grid is massively refined in the vicinity of the roughness patch. The regions of constant grid resolution around the patch are evidenced in Fig.  17, where thin lines indicate a change of grid resolution of the Cartesian mesh. The streamwise and normal variation of the grid is shown in Fig.  17a, whereas the variation in the $\zeta$ direction is sketched in Fig.  17b. Note that instead of the single elements of the roughness patch, the thick line depicts the extent of the complete patch. Furthermore, the capsule surface is sketched flat. The gray dashed line in Fig.  17a indicates the surface of the spherical forebody. The Cartesian cell length in the innermost region with the highest resolution is $\mathrm{\Delta }d=1.945\mu \mathrm{m}$, and it is doubled from region to region. An example of the change in resolution (scaled by a factor of 2) is given in Fig.  17a at the outermost frame. In total, the mesh contains $300\times {10}^6$ cells and $30\times {10}^6$ cells are clustered in the refined vicinity of the roughness patch.

The streamwise velocity deficit with respect to the smooth configuration $u^{\prime }$ downstream of the second and third rows of the elements, that is, at $\xi =-L/2$ and $\xi =L/2$, is shown in Fig.  18. In each figure, the staggered and the aligned configurations are shown in the left and right half plane, respectively. Differences between the configurations are restricted to the region close to the wall, that is, for $\eta <0.1\mathrm{mm}$. Downstream of the staggered second row in Fig.  18a, the highest velocity deficit occurs downstream of the elements slightly above the top of the element at $\eta \approx 30\mu \mathrm{m}$. Note that it is more intense in the aligned configuration. In the channel between the elements, higher velocities, that is, a lower velocity deficit, are evident in the aligned configuration. This statement also holds downstream of the identically aligned third row in Fig.  18b. Inside the patch, the difference in the streamwise velocity between the flow-channel and the postelement location is higher for the aligned configuration.

The spanwise velocity component normal to the symmetry plane is evidenced in Fig.  19 downstream of the roughness patch. Again, the staggered and aligned roughness elements are compared. The velocity component of the aligned configuration in the positive half-plane is multiplied by $-1$ to yield a better comparison. The thick line indicates $\overline{\nu }=0$. The overall pattern for both cases is identical, but the inward flow, that is, the bright peaks in Fig.  19, is stronger for the staggered configuration. This can be attributed to the inward flow generated by the sum of the elements acting as a finite patch. This discussion is resumed in the subsequent analysis in Sec.  V.B on the infinite extent of the rows of roughness elements located on the hemisphere.

B. Hemisphere Roughness Simulations by TUM

The TUM-DNS were carried out on a restricted 3D domain extracted on the hemisphere geometry for $\widehat{s}\in [74,116]\mathrm{mm}$. Details of the restricted domain and the roughness patch are shown in Fig.  5. The roughness patch of the hemisphere consists of five squared elements in the streamwise direction. The size of each element and the streamwise spacing are described in Sec.  V.A. The patch is centered at $\widehat{s}=77.4\mathrm{mm}$. Dirichlet boundary conditions are applied at the inflow, Riemann invariants are used at the outflow, and azimuthal-periodic boundary conditions are used in the spanwise direction. By using periodic boundary conditions, the domain can be limited in spanwise direction to one single roughness periodicity ( $L$). As a result, a considerable reduction of the domain size and, consequently, of computational cost is achieved. The grid of the restricted domain consists of about $29\times {10}^6$ points (1320 points in the streamwise and 220 in the wall-normal direction clustered at the roughness location and 100 points equally spaced in the spanwise direction).

1. Steady Simulations

For the case of $Re/l=6.25\times {10}^6/\mathrm{m}$, we compare the results with the ones presented in Sec.  V.A for the HLB capsule configuration. We found that the presence of the roughness on the hemisphere has a similar effect on the flow as in the case of the HLB capsule. In particular, because several roughness elements are present in the spanwise direction in the case of the HLB capsule, the flow in the vicinity of the symmetry plane presents the same periodicity features as in the case of the rough hemispherical geometry. The profiles of the streamwise velocity at three different positions are shown in Fig.  20. Profiles are extracted at spanwise coordinate $\zeta =0\mathrm{mm}$. The roughness patch is centered on $\xi =0\mathrm{mm}$, whereas $\xi =-0.5\mathrm{mm}$ and $\xi =0.5\mathrm{mm}$ refer to positions at 0.5 mm before and after the roughness patch, respectively. A good match can be observed at all three positions. Moreover, Fig.  21 shows a contour map for the spanwise component of the velocity in the $\zeta -\eta$ plane at the position $\xi =2.5L=0.5\mathrm{mm}$ downstream of the last roughness row. Contour lines are used for the capsule results and color shading for the hemisphere results. Contour level spacing is $0.3\mathrm{m}/\mathrm{s}$. For clarity, the dashed line shows the projection of the roughness elements. Also in this case, the spanwise velocity data for the hemisphere and the capsule are similar.

Results for the base flow at $Re/l=18\times {10}^6/\mathrm{m}$ are also shown in Fig.  20. Compared with the case with lower unit Reynolds number, a less-stable boundary layer is expected, and a higher roughness height–to–boundary-layer thickness ratio is found, in particular, $k/\delta =0.069$ and ${Re}_{\mathrm{kk}}=25$.

Further numerical studies on the hemispherical geometry (not shown here) have been undertaken to investigate the influence of the patch length in the streamwise direction. In fact, compared with the experiments in the HLB, the roughness in the present analysis has a smaller spatial extension. However, we found that further lengthening of the rough region in the streamwise direction, obtained by adding roughness elements upstream of the patch, has a negligible influence on the flow downstream of the patch. Therefore, no significant influence is expected on the stability properties of the wake developing downstream of the last roughness element.

2. Unsteady Simulations

For the case of $Re/l=18\times {10}^6/\mathrm{m}$, the base flow downstream of the roughness patch has been analyzed with the help of spatial two-dimensional linear stability analysis (LST-2D). The code used to perform LST-2D has already been validated and tested in the case of wake flow instability behind isolated roughness elements [ 17, 18]. No modal instability could be found in the boundary layer downstream of the roughness patch.

To investigate the presence of possible nonmodal instability mechanisms (i.e., transient growth), time-varying pressure disturbances are introduced at the inflow of the restricted domain (side 2 in Fig.  5a) and the development of unsteady disturbances is analyzed by means of unsteady DNS. The disturbance is defined as a superposition of five spatial modes with random amplitude $A_n$ and phase ${\varphi }_n$,

$$p^{\prime }(\zeta ,\eta )=c(\eta )\cdot \sum _{n=1}^5{A}_n\cos (\frac{2\pi n}{{\lambda }_{\zeta }}\zeta +{\varphi }_n)$$(16)

where ${\lambda }_{\zeta }$ equals the spanwise length of the domain at the inflow position. The function $c(\eta )=e^{-{(\eta /\delta )}^3}$, with $\delta$ being the boundary-layer thickness, guarantees that the perturbation vanishes outside the boundary layer. Experimental investigations at HLB have revealed that relevant frequencies over the rough-wall capsule for large Reynolds numbers lie in the range 100–300 kHz [ 16]. Based on this observation, three different frequencies are investigated in the present analysis: $f_1=167\mathrm{kHz}$, $f_2=250\mathrm{kHz}$, and $f_3=333\mathrm{kHz}$. The resulting inflow condition for the pressure at $\xi =-3.4\mathrm{mm}$ is given by:

$$p({\xi }_0,\zeta ,\eta ,t)=\overline{p}({\xi }_0,\zeta ,\eta )+p^{\prime }(\zeta ,\eta )\sum _{m=1}^3\cos (2\pi f_{m}t)$$(17)

where ${\xi }_0$ is the streamwise coordinate value of the inflow boundary and $\overline{p}({\xi }_0,\zeta ,\eta )$ is the pressure distribution of the steady base flow.

Simulations over a long enough time interval corresponding to multiple periods of the forcing field are needed for the transient effects to vanish and we verified the convergence of the spectrum across the entire simulation domain. The spatiotemporal analysis is conducted by performing a two-dimensional fast Fourier transform (FFT). Any flow variable $g(\xi ,\zeta ,\eta ,t)$ is thus decomposed into spanwise wavenumber-frequency spectra, $G_{m,n}(\xi ,\eta )$:

$$g(\xi ,\zeta ,\eta ,t)=\sum _{m=0}^{M-1}\sum _{n=0}^{N-1}{G}_{m,n}(\xi ,\eta )\exp [i2\pi (n\zeta /N+mt/M)]$$(18)

where $M$ and $N$ are the number of time and space samples, respectively. The amplitude $A_{m,n}(\xi )$ of the modes $(m,n)$ is defined as the maximum value of $|G_{m,n}(\xi ,\eta )|$ at the position $\xi$.

Figure  22 shows the amplitude of the perturbed modes with regard to the streamwise component of the velocity as well as the evolution of higher modes for the frequency $f_1$. Values are normalized with the edge velocity at the inflow. The origin of the streamwise coordinate, $\xi =0\mathrm{mm}$, is set on the center of the roughness patch. Qualitatively similar results are found for the other frequencies and they are not shown here.

As predicted by LST-2D analysis, no amplified modes are found for the analyzed frequencies. In addition, no evidence of possible transient growth could be found for the considered disturbances. Even though a rapid disturbance growth at the roughness location is evident for the modes $(\cdot ,4)$ and $(\cdot ,8)$, their contribution to the total disturbance energy is small.

The absolute value of the time Fourier transform of the streamwise velocity $|{\widehat{U}}_m(\xi ,\zeta ,\eta )|$ at different streamwise positions is shown in Fig.  23 for $f_1=167\mathrm{kHz}$. The absolute values are normalized on a scale of 0 to 1 and values below 0.2 are blanked out. The black isolines indicate the streamwise-velocity distribution, and the red line the projection of the roughness element. For clarity, the roughness height and the boundary-layer edge are marked in the figure. At the inflow, the maximum value of the disturbance is found at a height of about $\eta =0.07\mathrm{mm}$, corresponding to about $0.14\delta$. Further downstream, viscous effects are responsible for a strong damping of the disturbance in the region close to the wall and the disturbance maximum moves to about $\eta =0.2\mathrm{mm}$. As the roughness does not produce a significant perturbation at this height, no significant interaction can be observed with the incoming unsteady disturbance.