The Impact of Upstream Static Deformation on Flow Past a Cylinder/Flare

Abstract

Reynolds-averaged Navier–Stokes simulations are performed for supersonic turbulent flow over a cylinder/flare with upstream surface distortion representative of structural deformation induced via fluid–structural and fluid–thermal–structural behavior. Broad parametric analysis is carried out through the generation of Kriging-response surfaces from a database of general simulations. A posteriori simulations are then carried out at parametric combinations that correspond to extrema in the Kriging response surfaces to gain deeper insights into the interaction between the surface distortion and flow responses. Upstream distortions tend to decrease, rather than increase, the peak pressure and heat flux loads on the flare compared to an undeformed cylinder. Furthermore, decreases in these quantities reach up to O(10%) compared to up to O(1%) for increases. Integrated quantities over the flare are relatively insensitive to upstream distortion. The corner separation length is the most sensitive quantity to upstream distortion, with protrusions tending to increase the separation length and recessions reducing the separation length. Modifications in the separation length of up to 40% are observed. Reductions in peak loads tend to correspond to increases in the corner separation length. The movement of the surface distortion relative to the corner indicates a negligible impact beyond 1.5 distortion lengths from the corner, and the largest impact on the corner separation length occurs when distortion is directly adjacent. These results are an important step toward understanding and quantifying the impact of surface deformations on downstream components.

1. Introduction and Problem Statement

The impact of deformed surfaces, due to aerothermoelastic behavior, on downstream components is a key area of study in high-speed vehicle development [1,2,3,4,5,6]. Locally, deforming surfaces perturb the boundary layer through induced pressure gradients, including shocks and expansion fans, which can either instigate or modify local flow separation [7,8]. The impact of such surface deformations on downstream components is currently not well understood. The goal of this paper is to explore the effects of canonical surface deformations and develop a parametric model for surface loads. For concreteness, a cylinder/flare configuration at supersonic turbulent flow conditions is considered.

A large body of recent research has investigated fluid–structure interactions (FSI) in high-speed flow [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24]. However, most prior work has primarily focused on local impacts on the flowfield, with an emphasis on dynamic aeroelastic effects. Downstream effects are generally ignored. Two noteworthy sets of observations highlight the difficulty in assessing downstream effects. A study by Beberniss et al. [5] detected frequencies corresponding to structural motion in the pressure spectra downstream of a shock–boundary layer interaction (SBLI) over a compliant surface. However, a similar study by Peltier et al. [6] could not replicate this downstream effect of panel compliance. The root cause of the discrepancy was unclear, and it remains uncharacterized.

Static aeroelastic behaviors, like passive control bumps, are also pertinent to the downstream response [25,26,27,28,29]. These bumps help manage SBLI separation or modify the boundary layer characteristics to enhance downstream-component performance. Evidence also suggests that mean static deformations may have more lasting effects on turbulent boundary layers than dynamic deformations [30]. Additionally, both intentional modifications and unintentional structural changes, including permanent deformations and damaged panels, can significantly affect downstream components [4,30].

In the present study, a cylinder/flare platform [31,32,33,34,35,36,37,38,39] was selected. The presence of corner flow, and the associated separation, arising in this and related configurations, such as a double cone, adds the necessary complexity that exists in slender high-speed vehicles. Despite the attention given to flare geometries, studies on fluid–structural interactions within this context remain relatively sparse, though recent studies analyzing FSI phenomena embedded on a ramp [18,19] and upstream of a ramp [40,41] indicate an emerging consideration of this configuration for FSI research.

This paper seeks to characterize the impact of upstream surface distortions, characteristic of mean aerothermoelastic deformations typically seen in thermally or mechanically buckled thin-gauge skin panels, on downstream components. To this end, Reynolds-averaged Navier–Stokes (RANS) simulations were performed and benchmarked against experimental data and then incorporated into surrogate models using Kriging to develop trend-level studies via broad parametric analysis. Key parametric combinations of interest are identified and studied in detail using a posteriori RANS simulations.

2. Methodology

2.1. Problem Description

A representation of the cylinder/flare configuration and a corresponding mesh are shown in Figure 1. Different surface distortions were introduced upstream of the flare corner to generate various flow perturbations whose impact on the corner flow separation and attendant fluid response over the flare surface could be studied. Examples of the types of surface distortions implemented in this work are found in Figure 2, defined as convex or protrusions, concave or recessions, combined convex–concave or protrusion–recession, vice versa, and general deformations.

The computational domain is an axisymmetric plane set above the axisymmetric cylinder/flare. The wall of the cylinder/flare is set to a no-slip, isothermal condition at a temperature of 300 K (room temperature). The computational domain is set to be the same as the experimental domain identified by Chism [42,43], where the cylinder body does not have a leading nose body. Therefore, the left boundary is set to a constant, uniform inlet boundary, and the boundary layer is allowed to grow naturally. The top and right boundaries are set to far-field extrapolation.

The flow and overall distortion parameters were chosen to emulate experimental conditions from Section 2.3, and they are listed in

Table 1. Constant simulation inputs.

Variable	Value
$M_{\infty }$	7
Re (1/m)	1.3 × 105
$T_{\infty }$ (K)	51
Distortion length, L (mm)	58

and

Table 2. Simulation input parameters.

Lower Limit	Variable	Upper Limit
0	Distance from flare, ℓ (panel lengths)	2
−3.3	Amplitude mode 1, $A_1$ (mm)	3.3
−2.2	Amplitude mode 2, $A_1$ (mm)	2.2

. The distortions introduced upstream of the cylinder/flare were developed using mode shapes corresponding to a pinned–pinned 1D panel configuration:

$${\delta }_v\left(x\right)=\sum _{i=1}^m{A}_{i}sin\left(\frac{i\pi x}{L}\right)$$

(1)

where $A_i$ is the peak amplitude of the corresponding mode number, i is the mode number of the distortion, L is the length of the distortion, and m is the total desired number of mode shapes of the distortion. Surface distortions spanning modes 1 and 2 were considered, i.e., $m=2$. The distortion length was kept constant at $L=58$ mm, while $A_1$ and $A_2$ are the modal amplitude parameters that are varied according to Table 2. Distortions are set in the cylinder at variable distances from the flare corner, where the distance from the flare to the end of the distortion, in panel lengths, is ℓ. The distances span from zero panel lengths from the flare (abutting the corner) to the end of the distortion ($\mathcal{\ell }=0$) to two panel lengths from the flare to the end of the distortion ($\mathcal{\ell }=2$).

2.2. Fluid Dynamics Model

Flow predictions were generated using the compressible Reynolds-averaged Navier–Stokes (RANS) solutions through the DOD CREATE AV CFD software (version 12.8.1) Kestrel [44,45,46,47]. The solution method uses a second-order spatial scheme and a second-order Newton time integration scheme with advective temporal damping and three subiterations. Steady-state solutions were obtained with a starting Courant–Friedrichs–Lewy (CFL) number of 100, ramped to 1000. Grid selection considerations and convergence results are presented in Appendix A.1, while an iterative residual convergence assessment is provided in Appendix A.2. For all simulations, the turbulence intensity at the inlet and freestream was set to 0.1, and the eddy viscosity ratio was set to 0.01.

2.3. Companion Experimental Configurations and Comparisons

The simulation parameters are modeled after representative experiments conducted at the University of Tennessee Space Institute (UTSI) Mach 7 wind tunnel, which is a Ludwieg tube with a free jet test section. A detailed description and characterization of the tunnel are available from Gragston et al. [48]. Various deformed cylinder and cylinder/flare experiments were tested [42,43] at the flow conditions listed in

Table 3. Simulation/experiment parameters.

${\mathit{M}}_{\mathit{\infty }}$	$\mathit{R}\mathit{e}$	${\mathit{p}}_0$	${\mathit{T}}_0$	${\mathit{p}}_{\mathit{\infty }}$	${\mathit{T}}_{\mathit{\infty }}$	${\mathit{T}}_{\mathit{w}}$	${\mathit{\rho }}_{\mathit{\infty }}$	${\mathit{u}}_{\mathit{\infty }}$
	(1/m)	(Pa)	(K)	(Pa)	(K)	(K)	(kg/m3)	(m/s)
$7.0$	1.3 × 105	1.07 × 106	560	$257.8$	$51.9$	300	$0.0173$	1010

.

Since adiabatic wall temperatures are difficult to measure experimentally [42], for simplicity, a modified Stanton number using total temperature, $T_0$, was used instead of adiabatic wall temperature, $T_{aw}$. The modified Stanton number is defined in Equation (2), and it is compared to RANS predictions in Figure 3. The value $x=0$ corresponds to the location of the flare corner. In the simulations, two turbulence closure models were considered: the Spalart–Allmaras (SA) model and the Menter–Shear Stress Transport (SST) model. Although all models display an increase in heat transfer near the same location ($x=0$, the flare corner), the downstream predictions show significant differences. The SA model tends to underrepresent the peak Stanton number and subsequent decay. In contrast, the SST model aligns more closely with the experimental results on the flare wall, more accurately predicting the peak value location and magnitude. Based on this assessment, the Menter–SST model is used for turbulence closure in the remainder of this paper.

$$St=\frac{q}{c_p{\rho }_{\infty }{u}_{\infty }(T_0-T_w)}$$

(2)

Schlieren visualizations from the experiments, including those for an undeformed cylinder/flare and a convex, positive mode 1 shape ($A_1=3$ mm, $A_2=0$ mm) positioned 1-panel length upstream of the flare ($\mathcal{\ell }=1$), were compared with the simulation results. The experimental and numerical schlieren results are displayed in Figure 4. Figure 4a,b shows experimental schlieren results from Chism [42], and Figure 4c,d depicts these same results but darkened to show the low contrast in shock densities compared to the low densities within the tunnel. In all cases, the shock waves due to the flow turn at the flare corner were about 34 to 35 degrees. The shock angles between the experimental and simulated data deviated by one degree or less, corresponding to an error of less than 3%. Furthermore, the shock given off the wall distortions was both 12 degrees. These results indicate the suitability of the RANS simulations for the present purposes with this configuration and flow conditions for the trend-level analysis to follow.

2.4. Surrogate Modeling

Even though RANS simulations are relatively more economical compared to scale-resolving approaches, a full factorial study of the parameter space using RANS simulations represents a significant computational expense. The present work addresses this challenge through a surrogate-based approach. Surrogates emulate CFD simulations within defined input domains [49,50,51], and have proven successful in emulating steady CFD predictions across a variety of complex flow fields [23,24,52,53,54,55,56,57]. Among such methods, Kriging models represent a relatively simple, efficient, and effective method for generating surrogate models, and they were selected for use in the current study [49,58,59].

A Halton sequence [60] was implemented using SciPy [61], an open-source Python library, to generate low-discrepancy and quasi-random input parameters. This method was chosen since it provides flexibility to adapt the number of samples throughout the study and also guarantees better space-filling characteristics in lower dimensions [60]. The bounds considered for the operating conditions parameter space are listed in

Table 4. Kriging model inputs, ℓ, $A_1$, $A_2$.

Lower Limit	Variable	Upper Limit
0	Distance from flare, ℓ (panel lengths)	2
−3.3	Amplitude mode 1, $A_1$ (mm)	3.3
−2.2	Amplitude mode 2, $A_1$ (mm)	2.2

.

An important consideration when generating Kriging surrogates for flow responses is the potential for linked parameters. [62] This occurs when the flow indirectly responds to the input parameters through a linked equation, such as the total distortion in Equation (1). In such cases, the generality of the surrogate model is degraded if the sampling of the input parameters is carried out naively in an independent manner [62]. The use of hyperellipsoid sampling ensures a consistent and uniform sampling over the total distortion amplitude (Equation (1)), as described by Grier et al. [62].

The development of the surrogate models closely followed the process outlined by Korenyi-Both [59]. Insights into the history, applications, derivation, and algorithms of Kriging models were detailed by Krige [63], Matheron [64], Korenyi-Both [59], and Rasmussen and Nickisch [65]. The overarching framework for crafting a surrogate model is depicted in Figure 5. The process is as follows:

• Determine the input–output relationship: First, the correlation between the input and output must be established. For this study, the modal amplitudes of distortions and the location relative to the flare corner served as input parameters, and the pressure, heat flux, and skin friction coefficient profiles were the output functions.
• Define the sample space: The next step was to identify a sampling method (for example, Sobol or Halton sequence) and sample space that comprehensively spans the desired input parameters.
• Generate training data: Training data were then generated using the database of RANS solutions discussed in Section 2.2.
• Construct a surrogate model: After the training data were collated, the Kriging surrogate model was generated, as described in Appendix B.
• Prediction using the trained model: Response surfaces were constructed using the Kriging model to fill in the space between RANS simulations and develop a trend analysis.

The Kriging models in this work were developed and implemented with the SciKit-Learn package in Python [66], and they used SciPy [61] optimization methods. Two hundred RANS CFD cases were generated to train and validate the surrogate model, spanning the parameter space using a Halton sequence. The kernel selection is among the most important factors in model performance [65,67]; thus, four kernel architectures were considered in this study: Radial Basis Function (RBF), Matern, WhiteKernel, and ConstantKernel. The details of each kernel can be found in the work of Duvenaud [67]. Additionally, several combinations of these four kernels were tested; 80% of the data, randomly selected, were reserved for training, 10% for model testing, and the remaining 10% for final validation. The selection and convergence of the models are discussed in Appendix B.

3. Results

Trend analysis was carried out using Kriging response surfaces with a focus on the peak and integral values of the pressure, heat flux, and corner separation length. The interactions of the distortion with the corner separation, as well as the downstream influence and persistence of upstream perturbations, were explored. Key distortion shapes and locations were studied in depth using a posteriori RANS simulations at identified input parameter combinations to elucidate the underlying physical mechanisms.

3.1. Impact of Distortion Shape

A Kriging surrogate model was developed, designed with only the modal amplitude contributions ($A_1$ and $A_2$) as variables. In this case, the distance from the flare of the distortion was set to a constant value of one panel length ($\mathcal{\ell }=1$). Thus, the downstream effects of distortion shape and size were isolated. The Kriging model for this study, defined in Equation (3), with details on the selection and generation in Appendix B.1, was used to generate pressure, heat flux, and skin friction profiles over the span of the input parameter space (

Table 5. Kriging model inputs, $A_1$ and $A_2$.

Lower Limit	Variable	Upper Limit
−3.3	Amplitude mode 1, $A_1$ (mm)	3.3
−2.2	Amplitude mode 2, $A_1$ (mm)	2.2

); examples of these profiles are found in Appendix B.1. The peak values of the pressure and heat flux profiles on the flare, as well as the integral of the pressure and heat flux over the flare, and the separation length at the flare corner for the Kriging model predictions are shown in Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10. The Kriging prediction for the maximal value in these figures is denoted with a star, while the minimal values are denoted with stars and numbers. A posteriori RANS simulations were conducted at parametric combinations of modal amplitudes that corresponded to the maximal and minimal values from the Kriging response surfaces for each quantity.

The maximal and minimal values and modal contributions from both the Kriging and the subsequent RANS simulations, as well as the percentage differences from the undeformed case, are listed in

Table 6. Peak pressure maxima and minima predictions.

	Kriging Prediction		RANS Prediction
Configuration	Pressure Peak (psi)	% Difference from Flat	Pressure Peak (psi)	%Difference from Flat
Flat, $A_1=0$, $A_2=0$	0.9403	N/A	0.9403	N/A
Max Pressure Peak Case, $A_1=-0.756$, $A_2=0.735$	0.9488	+0.91%	0.9488	+0.91%
Min Pressure Peak Case 1, $A_1=2.914$, $A_2=-1.262$	0.8298	−11.8%	0.8179	−13%
Min Pressure Peak Case 2, $A_1=2.699$, $A_2=2.018$	0.8343	−11.3%	0.8148	−13.3%

,

Table 7. Maxima and minima pressure integral predictions.

	Kriging Prediction		RANS Prediction
Configuration	Pressure Integral over Flare (psi-in)	% Difference from Flat	Pressure Integral over Flare (psi-in)	% Difference from Flat
Flat, $A_1=0$, $A_2=0$	2.502	N/A	2.502	N/A
Max Pressure Integral Case, $A_1=-1.620$, $A_2=0.0$	2.507	+0.19%	2.507	+0.19%
Min Pressure Integral Case 1, $A_1=2.914$, $A_2=-1.262$	2.452	−2.0%	2.416	−3.5%
Min Pressure Integral Case 2, $A_1=2.699$, $A_2=2.018$	2.448	−2.2%	2.414	−3.5%

,

Table 8. Maxima and minima heat flux peak predictions.

	Kriging Prediction		RANS Prediction
Configuration	Peak Heat Flux (W/m2)	% Difference from Flat	Peak Heat Flux (W/m2)	% Difference from Flat
Flat, $A_1=0$, $A_2=0$	$6.67\times {10}^4$	N/A	$6.68\times {10}^4$	N/A
Max heat flux peak case, $A_1=-0.540$, $A_2=0.307$	$6.67\times {10}^4$	+0.56%	$6.71\times {10}^4$	+0.56%
Min heat flux peak case 1, $A_1=3.131$, $A_2=-2.118$	$5.89\times {10}^4$	−11.8%	$5.44\times {10}^4$	18.6%
Min heat flux peak case 2, $A_1=3.131$, $A_2=2.018$	$5.96\times {10}^4$	−10.7%	$5.63\times {10}^4$	15.7%

,

Table 9. Maxima and minima heat flux integral predictions.

	Kriging Prediction		RANS Prediction
Configuration	Integrated Heat Flux over Flare (W/m2-in)	% Difference from Flat	Integrated Heat Flux over Flare (W/m2-in)	% Difference from Flat
Flat, $A_1=0$, $A_2=0$	$1.457\times {10}^5$	N/A	$1.457\times {10}^5$	N/A
Max heat flux integral case, $A_1=0.164$, $A_2=0.107$	$1.460\times {10}^5$	+0.01%	$1.458\times {10}^5$	+0.01%
Min heat flux integral case 1, $A_1=3.131$, $A_2=-2.118$	$1.427\times {10}^5$	−2.1%	$1.409\times {10}^5$	−3.3%
Min heat flux integral case 2, $A_1=3.131$, $A_2=2.018$	$1.428\times {10}^5$	−2.0%	$1.409\times {10}^5$	−3.3%

and

Table 10. Maxima and minima separation length predictions.

	Kriging Prediction		RANS Prediction
Configuration	Separation Length (in)	% Difference from Flat	Separation Length (in)	% Difference from Flat
Flat, $A_1=0$, $A_2=0$	1.93	N/A	1.93	N/A
Min separation peak case, $A_1=-3.131$, $A_2=0.164$	1.70	−11.9%	1.70	−11.9%
Max separation peak case 1, $A_1=3.131$, $A_2=-1.405$	2.47	+27.7%	2.56	+32.6%
Max separation peak case 2, $A_1=3.131$, $A_2=1.0$	2.41	+24.6%	2.47	+27.7%

. Compression/expansion contours depicting the shock structure of the flow response are shown in Figure 11, while the streamlines of the separation corner variations are shown in Figure 12. For the former, $\overrightarrow{v}·▽p$, was used, which takes positive values at adverse pressure gradients (shocks) and negative values for expansions, and contours of pressure greater than $0.8$ psi are overlaid in pink. For the latter, streamlines are displayed to visualize corner separation/recirculation bubbles, and contours of temperature greater than 445 K are overlaid in pink. Finally, pressure and heat flux profiles on the cylinder/flare are shown in Figure 13 and Figure 14, generated using RANS simulations of distortions that were identified to cause maximal and minimal integrated pressure and heat flux via the Kriging model

$$0.000342+6.656·Matern(lengthscale=4.2,\nu =1.5)+WhiteKernel(noiselevel=0.103)$$

(3)

The predicted peak pressure values of the Kriging model as a function of the input distortion amplitudes are shown in Figure 6. A narrow band of upstream surface distortions led to increases in the peak surface pressure, albeit only marginally (∼1%). Generally, the introduction of upstream distortions decreased the peak pressure, with maximum reductions for the bounds considered of up to 12%. Furthermore, it is evident that the introduction of upstream distortion tends to reduce the pressure peak by up to 12%.

A comparison of the Kriging and RANS pressure peak predictions for the extrema conditions is listed in Table 6. The Kriging model predictions captured the peak pressure values to a 2% difference from the RANS predictions. The RANS-predicted shock structures generated via these distortions are shown in Figure 11. The pressure gradient associated with the flare–corner shock induced corner separation. The initial flow turn after separation resulted in a separation shock; in the nominal flat case, the peak pressure occurred where the separated shear layer and separation shock impinged on the flare. Distortions introduced an additional set of waves. These altered the pattern observed in the undeformed case, and they modified both the peak value and the width of the pressure profile. In Figure 11b, the maximal pressure peak case occurs when a strong distortion-induced compression wave impinges directly at the separation-induced shock impingement location. Conversely, the pressure peak minima case occurs when a strong expansion wave impinges near the separation-induced shock impingement location, reducing the peak pressure value, as seen in Figure 11c,d. The compression waves from the front half of the distortion in these cases impinged further downstream, introducing additional peaks similar in magnitude to the separation-induced shock-impingement peak.

The model-predicted pressure integrated over the flare is shown in Figure 7. The upper and lower bounds of the integrated pressure were predicted to vary by about 2%, indicating that the impact of upstream distortion on the pressure was muted in an integral sense compared to the peak pressure.

A comparison between the Kriging- and RANS-predicted pressure integrals, presented in Table 7, showed a less-than-2% difference in the modeled value from the RANS result. Pressure profiles of the RANS simulations at the Kriging model predicted integral maxima and minima, as shown in Figure 13. The pressure profile for the maximal pressure integral case exhibited a widening of the peak pressure, which was caused by the distortion-induced shock impingement near this location. Conversely, when the pressure integral minima occurred, the primary pressure peaks were reduced due to the impingement of expansion waves; however, secondary peaks occurred downstream due to the impingement of compression waves given off of the distortions. This redistribution of the peak value led to only minor differences in the total integrated pressure on the flare compared to the undeformed cylinder configuration.

The Kriging response surface for peak heat flux is shown in Figure 8. The upper and lower bounds varied by 12% of the maximum value. In this case, the maximal peak heat flux was located near the undeformed state. This indicates that almost any perturbation to the boundary layer in this upstream region augments the boundary layer and recirculation corner to reduce the relative peak off-body temperature and wall heat flux.

A modest disagreement in the Kriging-modeled heat flux peak compared to the RANS heat flux peak was observed in the minima cases, listed in Table 8. There was an approximately 8% difference between the Kriging and RANS predictions at these minima values. This is likely attributable to the fact that these minima values are situated on the domain corners, rather than within them, where Kriging performance is expected to degrade. The streamlines from the RANS simulations are compared in Figure 12. Note that the undeformed case, shown in Figure 12a, closely aligned with the maximal heat flux peak case, and slight distortion and streamline alteration achieved a peak, as shown in Figure 12b. In contrast, the peak temperature contours off-body are substantially widened in Figure 12c,d, and the peak heat fluxes are reduced, while secondary peaks in heat flux begin to appear downstream.

The response surface for integrated heat flux is shown in Figure 9. The upper and lower bounds varied by approximately 2% of the maximum value, similar to the integrated pressures. The global maximal integral heat flux was near the undeformed state, with the maxima at only a 0.01% increase in the heat flux integral compared to the undeformed case. This again indicates that perturbations to the boundary layer in this upstream region mostly cause reductions in the integral heat flux, albeit to a lesser extent than the peak values.

Heat flux integral comparisons between the Kriging and RANS predictions, in Table 9, indicate that, while the accuracy of the modeled heat flux peak values was sensitive to being on the corners of the domain, the integral values differed by only about 1%. Heat flux profiles of RANS simulations at the Kriging model predicting integral maxima and minima are shown in Figure 14. The heat flux profile for the maxima integral, Figure 14b, exhibited marginal changes in the heat flux profile compared to the undeformed example. The minima examples, Figure 14c,d, have reduced peak values of heat flux; however, the heat flux peak was seemingly redistributed further downstream. This ultimately resulted in only a minor influence on the total reduction in the heat flux integral.

Finally, predicted separation lengths from the Kriging model for each input value are shown in Figure 10. The boundary layer flow over deformed surfaces is affected due to pressure gradients from a variety of sources, either due to the compression or expansion waves from the flow turning at the initiation of the distortion or due to the subsequent surface curvature itself. The boundary layer undergoes a deceleration as it passes through compression shocks and over positive surface curvature, and it accelerates through expansions and over negative surface curvature. For a pinned panel configuration, the edges of the distortion show discontinuous changes in slope—leading to either compression or expansion waves at the edges of the distortion, with smoothly varying curvature within. The combination of discontinuous slope changes and smoothly varying curvatures causes changes in the shape and velocities of the boundary layer downstream, as is evident in the changes to the streamlines and the separation corner downstream of the distortions. The results in Figure 10 suggest that mode 1 amplitude ($A_1$) strongly influences the separation length, with a minor influence from the mode 2 amplitude ($A_2$). As the mode 1 amplitude decreases (indicating a more negative distortion), the separation length diminishes, while an increase in the mode 1 amplitude (signifying a more positive distortion) extends the separation length. In contrast, the mode 2 amplitude has a pronounced impact on the separation length when the mode 1 amplitude is negative but less so when the mode 1 amplitude is positive, with almost no impact for the largest mode 1 amplitudes.

A comparison of the predictions for the separation lengths from the Kriging model and RANS simulations is shown in Table 10. Here, a roughly 3% variation was observed in the maximal separation length cases between RANS and Kriging. The streamlines from the RANS simulations were analyzed, as shown in Figure 15. For the minimal-separation-length case, Figure 15b, the streamlines converged significantly, indicating a shrinking of the boundary layer, and they led to a boundary layer that was more resilient to separation due to pressure gradients. In Figure 15c,d, the streamlines immediately after the distortion are spread out compared to the undeformed case, and the separation bubble is enlarged. This streamline pattern indicates a thicker boundary layer with a higher shape factor, which is more likely to separate at the corner of the flare due to the shock-induced pressure gradient.

One of the main drivers of the alterations in the separation onset appears to be the type of topology of the distortions. Consider, in the top half of Figure 10 and Figure 15b, that for distortions with strong recessions, the separation length is reduced, which is accompanied by a convergence of streamlines and a shrinking of the boundary layer. This is contrasted with the bottom half of Figure 10 and Figure 15c,d, which show that, for distortions with strong protrusions, the separation length is increased and is accompanied by a widening of streamlines, indicating a larger boundary layer. For intermediate distortions that have both protrusion and recession, the streamlines have a shifting convergence and expansion, indicating a more complicated effect on the boundary layer.

Next, consider the aggregated results listed in

Table 11. RANS predicted values for each maximal/minimal condition identified from the Kriging response surfaces. The cells are color-coded, based on their percentage differences from the undeformed values. The color gradient transitions from red (indicating a positive % difference) to white (indicating a 0% difference) to green (indicating a negative % difference), scaled according to the maximum and minimum values of the table.

RANS Simulation Case	Pressure Peak % Change	Pressure Integral % Change	Heat Flux Peak % Change	Heat Flux Integral % Change	Separation Length % Change
Max pressure peak, $A_1=-0.756$, $A_2=0.735$	0.91	0.14	0.30	−0.18	−4.9
Min pressure peak, $A_1=2.699$, $A_2=2.018$	−13.3	−2.9	−13.8	−2.7	26.2
Max pressure integral, $A_1=-1.620$, $A_2=0.0$	0.45	0.19	−0.21	−0.17	−7.3
Min pressure integral, $A_1=3.131$, $A_2=2.018$	−12.6	−3.5	−15.7	−3.4	30.3
Max heat flux peak, $A_1=-0.540$, $A_2=0.307$	0.87	0.14	0.56	−0.01	−3.4
Min heat flux peak, $A_1=3.131$, $A_2=-2.118$	−10.3	−3.5	−18.6	−3.3	43
Max heat flux integral, $A_1=0.164$, $A_2=0.107$	−0.06	−0.03	−0.08	0.01	1.0
Min heat flux integral, $A_1=3.131$, $A_2=2.018$	−12.6	−3.5	−15.7	−3.4	30.3
Min separation length, $A_1=-3.131$, $A_2=0.164$	−0.47	0.07	−1.9	−1.0	−11.9
Max separation length, $A_1=3.131$, $A_2=-1.405$	−12.5	−2.4	−13.6	−2.0	32.6

, providing insights into how targeting specific maxima or minima impacts the outcomes of other parameters. The cells in Table 11 are color-coded, based on their percentage differences from the undeformed values. The color gradient transitions from red (indicating a positive % difference) to white (indicating a 0% difference) to green (indicating a negative % difference), scaled according to the maximum and minimum values of the table. There was a consistent increase in the separation lengths for all the minimal peak and integral cases. Specifically, achieving an 18% reduction in the peak heat corresponded to a 43% increase in the separation length at the corner. Additionally, the Kriging-model-predicted maximas and minimas were corroborated with the a posteriori RANS predictions at these sites, as the pressure and heat flux peak and integral maxima and minima values were consistently the maxima and minima of the RANS cases simulated. However, there was an anomaly in the separation length predictions, and the Kriging-predicted maximum separation length did not correspond to the RANS-simulated maximum separation length, which occurred for a distortion of $A_1=3.131$ and $A_2=-1.405$. This discrepancy can be traced back to the inherent limitation and reduction in the accuracy of surrogate models at domain boundaries and corners.

3.2. Impact of Distortion Location

The study’s analysis was extended to also consider the distance of upstream distortion from the flare corner, ℓ. Parametric bounds for this case are listed in Table 2. The Kriging models were generated from 300 additional RANS simulations. The generation and parameters of these new models are discussed in Appendix B.2. The selected model is defined in Equation (4):

$$0.0016+2.958·Matern(lengthscale=1.44,\nu =1.5)+WhiteKernel(noiselevel=0.0187)$$

(4)

For brevity, only the pressure peak and the corner separation length results are reported here. However, the trends for the heat flux peak, integrated heat flux, and integrated pressure were consistent with the trends observed herein for the pressure peak examples. Three-dimensional response surfaces were generated for this model, and in order to visualize them in two dimensions, one variable could be held constant while varying the other variables.

The predicted peak pressure values from the Kriging model, as a function of both the input distortion amplitudes and the location of the distortion, are shown in Figure 16. The x-axis corresponds to the length of the distortion from the flare, ℓ, and the y-axis is either the mode 1 or mode 2 amplitudes ($A_1$ or $A_2$). In Figure 16a, $A_2$ is held constant at 0, and in Figure 16b, $A_1$ is held constant at 0. From left to right on the contours, there are demarcations at approximately $\mathcal{\ell }=0.5$ and $\mathcal{\ell }=1.5$. In the first zone, near the flare corner where $0\le \mathcal{\ell }<0.5$, Figure 16a,b shows inverse patterns of each other. Negative mode 1 distortions were correlated with decreased peak pressures on the flare, as seen in the top left corner of Figure 16a, whereas positive mode 1 distortions resulted in increased peak pressures, seen in the bottom left corner of Figure 16a. In contrast, negative mode 2 distortions led to an increase in the pressure peak, seen in the top left corner of Figure 16b, whereas positive mode 2 distortions resulted in a decrease, seen in the bottom right corner of Figure 16b. This pattern suggests that positive, convex distortions near the flare corner (positive mode 1 and negative mode 2) increased the pressure peak, and negative, concave distortions near the flare corner (negative mode 1 and positive mode 2) decreased the pressure peak on the flare. As the distance from the flare corner of the distortion increases past $\mathcal{\ell }>0.5$, the responses were inverted, and the response surfaces reverted to the trends seen previously in Section 3.1 for $\mathcal{\ell }=1$. Finally, as the distortions were moved even further from the flare corner, beyond $\mathcal{\ell }>1.5$, the influence of upstream distortion on the flare peak pressure diminished. Note that the convex, positive mode 1 distortion ($A_1>0$) had the most lasting impact on the flare pressure peak values.

Deeper insights were gained using RANS simulations of flat ($A_1=0,A_2=0$), convex, positive mode 1 ($A_1=2,A_2=0$), concave, negative mode 1 ($A_1=-2,A_2=0$), combined convex–concave, positive mode 2 ($A_1=0,A_2=2$), and combined concave–convex, negative mode 2 ($A_1=0,A_2=-2$) distortions. Here $\mathcal{\ell }=0$ and $\mathcal{\ell }=2$. The resulting shock structures for distortions against the flare corner ($\mathcal{\ell }=0$) are depicted in Figure 17 and far from the corner ($\mathcal{\ell }=2$) in Figure 18. The introduction of the distortion within the corner separation region, Figure 17, was the root cause of the delineation seen at $\mathcal{\ell }\approx 0.5$ on the pressure peak response surface in Figure 16. In Figure 17b,e, the convex distortions are located within the recirculation, and they reduce the separation length. The distortions near the flare were accompanied by compression shock wave formations that impinged at the pre-existing pressure peak location. Conversely, when concave distortions were adjacent to the flare, and within the separated region, there was an expansion in the recirculation area, as seen in Figure 17c,d. In Figure 17c, the separation trails along most of the recession, with the initiation of the separated region near the beginning of the distortion. However, the flow turn angle due to the separation is milder, indicating a weaker shock that impinges on the flare and, consequently, a smaller peak pressure. Similarly, in Figure 17d, the separation region overlaps with the second half of the distortion. However, the turn angle dedicated to the separation is weaker than that of the turn angle at the start of the distortion. As such, the shock waves are spread out on the flare, leading to a smaller pressure rise than that caused by only the separation corner.

In Figure 18, the shock structures of the distortions located two panel lengths from the flare ($\mathcal{\ell }=2$) are shown. As expected from Figure 16, upstream distortions at two panel lengths from the flare ($\mathcal{\ell }=2$) negligibly affected the downstream peak pressures, as the corner separation shock wave was much stronger relative to the distortion-generated shock. Further, these secondary shocks dissipated before reaching the flare. The compression and expansion waves from the distortions caused pressure variations on the flare surface. However, the impact was notably less pronounced compared to distortions closer to the flare or distortions that generated shock impingements at the same location as the corner separation shock. The dissipation and weakened responses caused by distortion-induced shock waves were the underlying causes of the reduced responses for distortions far from the flare ($\mathcal{\ell }>1.5$).

The predicted corner separation length using the Kriging model is shown in Figure 19, where the mode 2 amplitude, $A_2$, is held constant at 0 in Figure 19a, and the mode 1 amplitude, $A_1$, is held constant at 0 in Figure 19b. Three different responses appeared when varying ℓ and $A_1$, as shown in Figure 19a: the first ranging from $0\le \mathcal{\ell }<0.3$ for distortions against the flare; a second extending from $0.3<\mathcal{\ell }<1.5$; and a third for distortions far from the flare for $\mathcal{\ell }>1.5$. However, only two responses occurred when varying ℓ and $A_2$, as shown in Figure 19b: the first for distortions near the flare, $0\le \mathcal{\ell }<0.6$, and the other for distortions away from the flare, $\mathcal{\ell }>0.6$. For both, a pattern emerged when the distortion was adjacent to the flare and situated within the corner separation. As convex distortions approached the corner, the separation length was reduced, and as concave distortions approached the corner, the separation length increased. The underlying cause of this effect is apparent from analyzing the responses of the streamlines for the distortions near the corner, shown in Figure 20. The streamlines demonstrate that convex distortions near the flare significantly reduced the overall separation length, as shown in Figure 20b,e. In contrast, concave distortions extended the separation further upstream, as shown in Figure 20c,d. These behaviors were expected, as concave/convex surfaces introduce pressure gradients on the boundary layer, affecting skin friction and separation [7,27,68], which are the basic principles of shock control bumps [69,70,71,72]. As the distortions shifted beyond approximately $\mathcal{\ell }>0.6$, the trends in the separation length shifted towards the results discussed in Section 3.1 and shown in Figure 6 for $\mathcal{\ell }=1$. Finally, for distortions beyond $\mathcal{\ell }>1.5$ in Figure 19a, a rapid return to the undeformed cylinder/flare separation length occurred. The restoration of the response to the undeformed separation length occurred faster for concave distortions compared to convex distortions where the separation length was restored at a slower rate. Note, in Figure 21, that the streamlines reverted to the nominal state approximately halfway between the distortion and the separation region, confirming the diminished influence of the distortion on the downstream flow as the distance from the separation and flare corner increased beyond $\mathcal{\ell }=0.6$, as predicted in Figure 19b. The concave, negative mode 1 distortion response returned to undisturbed streamlines quickly following the distortion, as shown in Figure 21c. However, the convex, positive mode 1 distortion had a more lasting effect on the streamlines, as shown in Figure 21b. Furthermore, for the mode 2 distortions, Figure 21d,e, the streamlines returned to their nominal state quicker compared to the mode 1 distortions. This is reflected in Figure 19b, as the separation length contour beyond $\mathcal{\ell }=0.6$ shows little change compared to the undeformed case for any distortion at any location.

4. Concluding Remarks

An approach to parametrically characterizing the impact of static surface distortions on a downstream cylinder/flare configuration was developed. The distortions are representative of structural deformations induced via fluid–structural and fluid–thermal–structural interactions. A trend analysis was carried out using RANS simulations in conjunction with Kriging response surfaces. Analyses of Kriging-model-generated response surfaces were conducted by varying the modal amplitudes of the distortion and the location of the distortion with respect to the cylinder/flare. Attention was focused on the peak and integral values of the pressure and heat flux, as well as changes in the cylinder/flare corner separation length. The Kriging model was used to identify key trends and parametric combinations of interest. The phenomena contributing to the trends and predictions were affirmed through a posteriori RANS simulations of parametric combinations, selected from maximal/minimal conditions in the different model response surfaces.

These analyses yielded the following conclusions:

• The introduction of upstream surface distortion tended to decrease peak pressure and heat flux values, in extreme cases by O(10%). However, integrated pressure and heat flux were only marginally impacted (1%). This behavior was attributed to reductions in primary peaks and the introduction of secondary peaks.
• The flare corner separation length was highly sensitive to upstream distortions, decreasing by up to 12% and increasing by upwards of 43% compared to the undeformed cylinder/flare.
• RANS characterizations highlighted several marked impacts of upstream distortion on the downstream flow response. The local compression and expansion surfaces give rise to shocks and expansion waves, which impinge on and affect the loading on the flare surface by augmenting the primary peaks at the separation shock impingement, as well as introducing additional peaks further downstream. Upstream deformation-induced perturbations to the boundary layer that influence the downstream separation corner and reattachment behavior affect the separation length and heat flux profiles.
• Single protrusions into the flow predominantly lead to longer separation lengths at the corner, while a single recession tends to reduce it. Upstream distortions cause varying degrees of alteration to the boundary-layer state, evident in the convergence and expansion of the streamlines of the flow and demonstrating the ability of the distortion to affect the boundary-layer state as it travels to the flare corner and influences separation behavior.
• The output functions of interest show some correlation behavior, as decreases in the peak and integrated values of pressure and heat flux correspond with large increases in separation lengths. Conversely, decreases in separation lengths correspond to only small changes in the peak and integrated values of pressure and heat flux.
• Distortions within the separation region ($\mathcal{\ell }<0.5$) cause distinct flow responses compared to distortions outside of the separated region. Convex distortions reduce the separation size, while concave distortions extend the separation upstream.
• The impact of distortion diminishes rapidly when placed approximately 1.5 panel lengths ($\mathcal{\ell }>1.5$) or farther from the flare. Distortions protruding into the flow tend to cause more lasting effects, while recessed distortions cause the flow to return to a nominal state sooner. Mode 2 distortions with combined protruded and recessed distortions cause less of an impact downstream overall compared to the mode 1 cases when far from the flare.

Overall, the results and conclusions of this study elucidate a variety of roles that upstream surface distortions play on downstream flow responses both in the projected shocks and expansions and in boundary-layer alterations.