Keywords

1 Introduction

Shock-wave/boundary-layer interactions [1] occurring within a supersonic flow can greatly affect the performance of a vehicle or a propulsion system due to the presence of large recirculation regions, intense local heating and a loss of efficiency of the aerodynamic control surfaces. An important example of shock/boundary-layer interaction occurs when the slope of the body surface suddenly changes inducing a sharp compression of the flow near the wall: as a consequence, the compression waves coalesce within the boundary layer into a shock. The prototype of this type of flow is the supersonic/hypersonic flow past a compression ramp that has been extensively studied by various researchers over the last decades.

Accurate skin-friction and wall heating predictions require the (possibly accurate) modelling of the shock-wave: a task which can be achieved using either shock-capturing (S-C) or shock-fitting (S-F). Shock-capturing methods, which are nowadays very popular owing to their algorithmic simplicity, are built upon the integral, conservation-law form of the governing equations. Despite their great popularity, shock-capturing methods are known to be plagued by a variety of numerical problems that often undermine the reliability of the computed solutions. Modelling shock-waves via shock-fitting consists in identifying and tracking the motion of the discontinuities, which are treated as internal boundaries of zero thickness which bound regions of the flow-field where a continuous solution of the governing PDE exists. The motion of the discontinuity and its upstream and downstream states are computed by applying the Rankine-Hugoniot (R-H) jump relations whereas the flow in the smooth regions can be approximated using any available gas-dynamic solver. Even though the shock-fitting techniques are immune to most of the numerical troubles incurred by shock-capturing methods, their use is nowadays very limited, primarily because they are considered algorithmically complex and only applicable to “topologically simple” flows. Indeed, the shock-fitting techniques which were very actively developed by Moretti [2] and his collaborators starting in the 1960s relied on structured meshes, which required careful coding in order to deal with complex shock-patterns. By the late 1980s, only an handful of research groups worldwide was still active in the development and application of shock-fitting techniques (on structured meshes). Over the last ten years, however, some of the authors [3, 4] have developed a novel unstructured shock-fitting technique which allows to relieve many of the algorithmic complexities that plagued the “traditional” shock-fitting techniques developed on structured meshes. This paper describes an updated version of the unstructured shock-fitting algorithm, which allows to compute shock-wave/boundary-layer interactions. The algorithm is applied to the computation of hypersonic compression ramp flows and the results obtained using both shock-capturing and shock-fitting simulations on nearly identical triangular meshes are compared to the experimental data in order to highlight the differences between the two shock-modelling approaches.

2 Shock Fitting Algorithm

A shock-fitting technique for unstructured grids was proposed for the first time by some of the authors in [4] and later improved and enhanced in [5,6,7]. Even though the technique has already been used [8, 9] to simulate hypersonic viscous flows, those flow-cases did not involve shock/boundary-layer interactions, which require the additional algorithmic ingredients to be described in this paper.

Firstly, the presence of highly elongated triangular cells, such as those commonly found in boundary-layer meshes, puts a further constraint on the mesh generation algorithm that is used to locally re-construct the grid around the moving shock. Secondly, new modelling options were introduced to compute the shock-slope within the end-points of the fitted-shocks that fall inside the boundary layer.

In order to describe this improved version of the shock-fitting algorithm, we start by describing how data is stored. Figure 1a is what we call the “background” mesh: a triangulation of the entire computational domain which will also be used to perform the shock-capturing calculations presented in Sect. 4. The fitted shock, which is shown all frames of Fig. 1 using a red line, is made up of an ordered sequence of shock-points, which are mutually connected using straight shock-edges. Both the shock-points and shock-edges are duplicated items that share the same geometrical location: the shock-upstream and shock-downstream flow states along the fitted shock are stored within each pair of shock-points and by duplicating the shock-edges the fitted-shock behaves like an interior boundary of zero thickness. Observe from Fig. 1a that the geometrical location of the shock-points is totally un-related to the location of the grid-points of the background triangulation. Let us now suppose that the solution at time t is known within all the grid-points of the background mesh and all pairs of shock-points. The algorithm that updates the field solution and the shock states and position at the time \(t+\Delta t\) consists in five steps, which will be described hereafter.

2.1 Step 1: Cell Removal Around the Shock Front

The first step consists in removing those triangles of the background mesh which are crossed by the discontinuity, and also the triangular cells that have at least one of their vertices that is too close to the shock-front. By doing so, a mesh-less hole, which contains the discontinuity, is carved within the background mesh, as shown in Fig. 1b.

2.2 Step 2: Local Re-meshing Around the Shock Front

The hole is separately re-meshed using a constrained Delaunay triangulation (CDT), see Fig. 1c: the shock-segments and the boundary-edges of the hole are constrained to belong to the final CDT and no further grid-point is added. The re-meshing process is localized around the discontinuity and, therefore, it has a limited computational cost.

2.3 Step 3: Grid Assembly

The “computational” mesh that will be used to advance the solution in time from t to \(t+\Delta t\) is made up of the two different grids that have been generated in the previous two steps, see Fig. 1d: the background mesh with the hole and the CDT that fills the hole. When merging these two meshes, it is important to remove the duplicated nodes and ensure a consistent numbering of both the grid-points and the triangular cells.

2.4 Step 4: Calculation of the Unit Vectors Normal to the Shock Front

The tangent and normal unit vectors along the shock-front are needed within each shock-point in order to apply the R-H relations (step 5). These unit vectors are computed using finite-difference formulae which involve the coordinates of the shock-point itself and those of its neighboring shock-points. Depending on the local flow regime, it may be necessary to use upwind-biased formulae to avoid the appearance of geometrical instabilities along the fitted discontinuities.

Fig. 1.
figure 1

Sequence of steps for the generation of the computational mesh at time t

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The presence of the boundary layer introduces a substantial difference with respect to the inviscid case. Indeed, in the inviscid case, the flow adjacent to a solid surface (wall) can be supersonic, so that shock waves may eventually reach the wall, in which case the slope of the shock at the wall depends upon the local curvature of the wall. This is not any longer true in the viscous case, because the presence of the boundary layer prevents the shock wave from reaching the wall. It follows that in the viscous case the first shock-point lies (within the supersonic region of the boundary layer) at some distance from the wall, and, therefore, it is not any longer possible to use the wall geometry to compute the shock-slope. It should also be avoided to use the neighboring shock-points, since this would imply violating the domains of dependence.

Before describing how we compute the normal to the shock within the first shock-point inside the boundary layer, we need to distinguish two possible situations: (i) the shock is formed within the supersonic zone of the boundary layer through the coalescence of compression waves and it emerges from the boundary layer as an oblique shock; (ii) an oblique or quasi-normal shock, formed outside the boundary layer, penetrates into the boundary layer decreasing its intensity and disappearing as soon as it reaches the sonic line.

In the former case, because the intensity of the shock wave in the first shock-point is certainly very small, it can be assumed as in [10] that the local slope is that of the corresponding Mach line upstream of the shock and, therefore, the vector normal to the shock can be taken perpendicular to the Mach line, as shown in see Fig. 2a. Also in the latter case the local slope in the end-point of the first shock-point can be assumed to be parallel to the Mach line upstream of the shock, but in this specific case, since the point shock-point is on the sonic line, the normal vector is parallel to the velocity vector (the Mach angle equals \(\pi /2\)), see Fig. 2b.

Fig. 2.
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Special points modelling

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2.5 Step 5: Solution Update Using a Shock-Capturing Code and Enforcement of the R-H Relations

Using the computational grid as input, a single time-step calculation is performed using an unstructured, vertex-centered, shock-capturing solver [11], which returns updated nodal values at time \(t+\Delta t\). As explained in details elsewhere [4], the dependent variables within the shock-points located on the downstream side of the shock need to be corrected by enforcing the R-H relations across each pair of shock-points. This amounts to solve a system of non-linear algebraic equations which also supplies the local shock-speed. Finally, knowledge of the shock-speed within all shock-points allows to move the fitted-shock-front to its updated location at time \(t+\Delta t\) so that solution can be advanced over the next time-interval by repeating steps 1 to 5.

3 Numerical Simulation of Shock/Boundary-Layer Interaction

In this section we will analyze and discuss the results obtained in the simulation of the hypersonic flow past two different compression ramps. These flow configurations have been experimentally studied by Holden [12]. Table 1 shows the flow conditions examined herein, which only differ in the ramp deflection angle, \(\alpha \). Both flows are assumed to be laminar, in agreement with the experimental evidence [12].

Table 1. Shock/boundary-layer interaction: flow conditions.
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The two flow conditions were simulated by means of the in-house S-C solver described in [11], which is the same solver used in the smooth regions by the S-F technique. Then, the approximate location of the fitted-shocks was extracted from the S-C solutions in order to initialize the S-F computations: since no shock-detection algorithm is currently implemented, this procedure is manually performed. Moreover, the coupling of the shock-fitting technique with a shock-capturing solver enables an hybrid mode of operation, in which some of the shocks are fitted (in this case we chose the strongest ones) whereas the remaining ones are captured.

3.1 Pressure Field Analysis

Before analyzing the numerical results, it is instructive to examine how the shock pattern is affected by the selected ramp deflection angle. The pressure fields computed by means of S-F are shown in Fig. 3, resp. Fig. 4, for the \(\alpha = 15^{\circ }\), resp. \(\alpha = 24^{\circ }\), ramp deflection angle. In both cases, a relatively weak shock is formed at the leading-edge (LE).

In the \(\alpha = 15^{\circ }\) case, the LE shock collides with the coalescence shock and their interaction gives rise to a resulting shock, a shear layer and an expansion fan that interacts with the boundary-layer that develops along the ramp.

Fig. 3.
figure 3

Pressure field: \(15^\circ \) case.

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Fig. 4.
figure 4

Pressure field: \(24^\circ \) case.

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Flow separation occurs when the ramp deflection angle is increased to \(24^\circ \), which is the case shown in Fig. 4. The adverse pressure gradient leads to the generation of a separation bubble, whose dimensions depend on both the Mach and Reynolds numbers. The presence of the separation bubble causes a deviation of the streamlines and leads to the formation of a separation shock due to the coalescence of compression waves. The recirculation bubble terminates along the ramp in the re-attachment point, where a re-attachment shock is generated again due to the coalescence of compression waves. The interaction between these shocks generates an even stronger resulting shock, a shear layer and an expansion fan.

3.2 Numerical Models

The two selected test-cases were computed by the S-F and S-C solvers using two different grids that are highly stretched in the wall-normal direction; their features, which include the number of cells, minimum and maximum wall-normal spacing and longitudinal spacing, have been reported in Table 2. In order to properly resolve the boundary layer, the height of the first cell at the wall has been set equal to \(2\cdot 10^{-4}\). The unstructured meshes were obtained from structured ones by subdividing each quadrilateral into two triangles using one of the two diagonals. Therefore, the unstructured mesh has the same number of nodes of the original structured mesh, but twice as many (triangular) cells.

Table 2. Spatial discretization
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4 Results and Discussion

The dimensionless wall pressure and skin friction distributions corresponding to the different ramp deflection angles are plotted in Figs. 5 and 6: for comparison, the experimental data by Holden [12] are also reported. Each plot shows the results obtained using both S-C and S-F on two grids that are identical except in the regions where the fitted shocks are located.

Fig. 5.
figure 5

Wall pressure distribution: comparison with experimental data

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Fig. 6.
figure 6

Skin friction distribution: comparison with experimental data

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In the attached flow case (\(\alpha =15^{\circ }\)) the coalescence shock has been fitted, whereas the LE shock has been captured. By doing so, there are no noticeable differences between the wall pressure distributions obtained by means of S-C and S-F. Both numerical results are in fairly good agreement with the data supplied by Holden [12], even though the computed values are slightly higher than those found in the experiment. This could be due, however, to the three-dimensional effects due to the finite span of the ramp, which cannot be accounted for in a two-dimensional simulation.

Fig. 7.
figure 7

Triple-point region: comparison between the captured and fitted solutions.

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The negative skin friction values in Fig. 6 show that flow separation takes place when the wedge angle is increased to \(\alpha =24^{\circ }\). In this second test-case, we have chosen to fit the separation and transmitted shocks, whereas the re-attachment shock has been captured. The extent of the separation bubble is well predicted by the S-F technique, whereas the S-C calculation estimates a much larger bubble, probably because of the increased numerical diffusion. Figure 7 highlights the reason behind the overestimate: when the separation shock is captured, its thickness is comparable to the distance between the triple-point and the wall. This causes a significant widening of the triple-point region, that affects the position of the re-attachment point and, consequently, also the position of the separation point. It follows that the size of the separation bubble in the S-C simulation is much larger than the one predicted by S-F or experimentally measured. Moreover, the pressure peak, due to the triple-point shock interaction, is slightly shifted with respect to the experimental data. The S-F technique predicts well also the extent of the high-pressure region. To summarize, the quality of the S-F solution is improved because, by reducing to zero the thickness of the separation shock, the extent of the triple-point region is brought closer to its “true” (physical) size. An even better result can be expected, if all the discontinuities that meet at the triple point were fitted. On the other hand, a similar result could only be obtained in the S-C calculation by repeatedly refining the mesh, thus incurring in a significantly higher computational cost. Indeed, anisotropic mesh adaptation requires a precise knowledge of the shock location. However, since the position of the shocks is not known a priori, the calculation of a S-C solution featuring the correct position of the shock waves requires an iterative process, consisting in the generation of a sequence of anisotropically adapted meshes and the corresponding set of flow simulations.

5 Conclusion

An existing shock-fitting technique for unstructured grids has been further developed and applied for the first time to the simulation of laminar shock/boundary-layer interactions, in particular those involving the hypersonic flow over compression ramps. Even though the shock-fitting simulations were obtained using an hybrid approach, whereby only some of the shocks were fitted and the remaining ones were captured, the comparison between the hybrid approach and a fully captured simulation has clearly shown the advantages of the former. This promising result prompts us to proceed with further testing which may include: oblique shock reflections, shock/boundary-layer interactions in the turbulent flow regime and transonic flows along profiles.