Computational analysis of shock wave diffraction for convex slabs

Abstract

The present study is centered on the vortexlets in the shock wave diffraction over three different slabs (60°, 90°, and 120°) for shock Mach numbers of 1.65, and 3.0. The third-order accurate implicit solver is built on advection upstream splitting along with least squares cell-based method and utilizes the benefits of refined mesh in the regions having high discontinuities. Vortexlet formation, pressure ratio and specific heat flux on the step wall, and movement of the separation point are some of the key aspects of the present analysis. For the numerical simulation of the moving shock, the Finite Volume Method is utilized to find the solutions of the governing equations. Vortexlets, secondary shock, embedded shock, contact surface, slipstream, expansion fan, and vortex are captured precisely. Apart from isopycnics; isobars, isotherms, and velocity contours are plotted as well. Our results emphasize the fact that there exists two types of vortexlets, which are different in their positions apart from their driving mechanisms.

Introduction

Shock wave diffraction is a phenomenon that comprises reflection of the shock and deflection of the flow. The most common example is an explosion or detonation wave. For shocks moving past a change in the geometry, we can predict and mitigate the damage level of the blast waves to a certain limit. Defense research, aeronautical design are some of the practical applications of shock diffraction study.

Figure 1 represents schematic of various features like Incident Shock (IS), Upper Diffracted Shock Wave (UDSW), Last Running Expansion Wave (LREW), slipstream or Shear Layer (SL), Primary Vortex (PV), Secondary Vortex (SV), Lower Diffracted Expansion Wave (LDEW), Contact Surface (CS) and Embedded Shock (ES) as well. The ratio of the velocities of the fluid and the sound in that medium is represented by a non-dimensional parameter called Mach number. M_s is used to denote incident shock Mach number in the present study. The diffracted shock waves from different corners at a particular M_s create an envelope named as main shock curve. However, the demarcation is done by the wall shock, not being a part of the envelope and meeting the wall perpendicularly. The Last Running Expansion Wave (LREW) is the border of a sound disturbance traversing into the post-shock fluid. When the post-shock flow is subsonic, this boundary becomes a circular arc; the center of which is the disturbance source moving with lesser speed than that of sound (the radius). That is why there is upstream movement of disturbance in this case. However, for supersonic post-shock flow, the front of the expansion wave becomes tangent to the Prandtl–Meyer expansion line. The flow is accelerated by the expansion waves and becomes parallel to the separation streamline or Shear Layer (SL), which in turn rolls backward to form the Primary Vortex (PV). Afterward, the secondary wave shocks the flow. The rearward-facing secondary shock [Here referred to as Shock Wave (SW) in general] is the connector for the pressure difference between high-velocity, low-pressure rarefied gas over the slipstream and low-velocity, high-pressure gas behind the diffracted shock. Whereas, the flow fields governed by the incident and diffracted shocks are demarcated by the Contact Surface (CS) or vortex sheet. The small vortices around PV and at the trailing edge of SL are referred as vortexlets.

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Fig. 1

Schematic of characteristics produced by shock wave diffraction over 90° slab for M_s = 1.65

Figure 1 specifically presents the case of shock diffraction over 90° slab for subsonic post-shock flow. The Incident Shock (IS) being diffracted from the upper or first corner eventually becomes the Upper Diffracted Shock Wave (UDSW), which subsequently appears as incident normal shock relative to the diffraction wall for the lower or second corner. The shock curvature is changed accordingly. The shock diffraction from the lower corner also generates the Secondary Vortex (SV) and the Lower Diffracted Expansion Wave (LDEW). However, the span of SV is much smaller than the PV generated from the diffraction over the first corner. LDEW meets UDSW at the same point where the shock curvature changes. ES is associated with the PV for the accelerating flow along with the vortices. The geometrical dimensions shown are all in S.I. units, i.e., meters.

A lot of numerical, experimental, and analytical studies were performed to visualize the different flow characteristics in perturbed regions.

The fundamental physical descriptions needed to understand the compressible flow were given by Anderson [1]. Banerjee and Halder [2] highlighted the separation and reattachment, slipstream angle variation with corner angle and M_s, pressure load and specific heat flux variation along the step wall, etc., by numerical simulations using Euler model. Comparison of primary and secondary vortices while analyzing separation in upstream and downstream walls of a round corner was done by Banerjee and Halder [3] using Navier–Stokes modeling. Bazhenova et al. [4] examined the shock wave interactions with both of concave and convex surfaces. Ratios of pressures and heat fluxes were plotted along with M_s at the tail of the expansion waves and at the wall shocks. Regular and irregular reflections of shock waves were reported by Ben-Dor [5]. Brahmi et al. [6] observed the increment of the transition angles with M_s. At M_s = 4.5, the shifting of a Single Triple Point (STP) into a Double Triple Point (DTP) and again back to STP occurred at the end of the second reflector, which indicates the flow is retaining the memories. STP and DTP are generated by irregular or Mach reflections of shocks. The three parts of STP are incident, reflected shocks, and the Mach Stem. The first Mach Stem eventually becomes the IS for the second reflection in DTP. Brahmi et al. [7] proved that for the vorticity transport equation, the vortex stretching due to the compressibility effects is prominent for M_s ≤ 2.5, where viscosity is dominant in the vorticity diffusion in comparison with baroclinic effects. Whereas, this trend is reversed for M_s ≥ 3.5. The characterization of a blast wave was studied by Gavart et al. [8]. Obstacles with different diameters or materials and reason for the difference in sensor measurements were not covered in this literature. Gnani et al. [9] investigated shock wave diffraction pattern over wedge and round-type geometries using Schlieren technique. Some of the features (e.g., core of the vortex) were found pretty much blurred in experimental results.

Numerical simulations were performed by Hillier and Netterfield [10] using the Euler and the Navier Stokes schemes over an edge of a step to study the separation point movement along the wall. Although, viscous effects for smaller corners were not mentioned in that paper. Computational analysis by Hillier [11] proved that the slipstream would be rolling back to create a vortex. He identified a pair of ES near the core of PV for 90° corner. Various flow features like vortex trajectory, secondary shock velocity, wall shock Mach numbers, etc., were calculated using the Euler model. However, viscosity should be accounted for to capture vortices in the case of lower M_s and for shock–vortex interactions.

Kofoglu et al. [12] analyzed the mechanism of formation of the vortexlets by utilizing a 2-D high-order Euler solver. They considered M_s = 1.3, 1.5, 1.7, and 2.0 and found two types of vortexlets at different locations and time intervals. However, the corresponding variation with different corner angles was not reported. Law et al. [13] investigated the oscillation of the separation point over convex curved walls behind diffracted shocks at large time and length scales. However, the transition M_s value where this motion would be converted to always downward was not found. Liou et al. [14] provided the details of AUSM (Advection Upstream Splitting Method) scheme. AUSM is used for splitting of convective and pressure fluxes. It is an amalgamation of Flux Difference Splitting (FDS) and Flux Vector Splitting (FVS) schemes widely used as techniques to capture shock waves. FDS is more accurate and takes a longer time. FVS is robust and fast technique. However, the numerical dissipation associated with this scheme is somewhat considerable. Experiments were performed for studying the effects of a flat wall at the exit of a rectangular nozzle on the shock structure and transverse deflection pattern of the jet by Manikanta and Sridhar [15]. Oliveira and Azevedo [16] used five different finite difference schemes to avoid the numerical dissipation in shock wave resolution.

Skews et al. [17] analyzed the behavior of the shear layer by conducting large-scale experiments in a shock tube for a limited range of M_s. It was evident that the separation and flow development is different in realistic scale studies due to instabilities, turbulent patches and boundary layer effects. The upstream flow would be of great importance in such cases. Skews [18] focused on diffracted shock profiles in detail on circular arc and multi-facetted corners. Although, separation of flow was not analyzed in details. Skews [19] performed experiments on shock wave diffraction over different corner angles and M_s ranging from 1.0 to 5.0. It was observed that the position of the slipstream, expansion wave, and the velocities of secondary shock, contact surface were independent of larger (> 75°) corner angles. Fair agreements were observed with theoretical results. Skews [20] conducted experimental studies also for prediction of shape of a diffracting shock wave for step angles ranging from 15 to 165° (in multiples of 15°) and M_s = 1.0 to 4.0. The results were validated with Whitham’s theory. The agreement was good for M_s = 3.0 but fair for other M_s values.

Sun and Takayama [21] pinpointed their analysis on small vortical structures formed near the core of PV by simulations using Euler and Navier–Stokes modeling. The results were compared with using Turbulence models and the small vortices were found to be suppressed in that case. Sun and Takayama [22] found several factors for the secondary shock formation by shock tube experiments and numerical simulations utilizing Euler solver. Their observations revealed that the effects of viscosity on the strength of the secondary shock are almost negligible. One other important finding about this paper was finding the limiting M_s = 1.346 for the formation of the secondary shocks in case of di-atomic gases. However, they did not represent relatively smaller corner angles, for which the slip lines would appear in a proximity to the step wall. Sun and Takayama [23] calculated circulation strengths of the vortices generated in shock diffraction for different step angles and M_s using self-similar Eulerian solutions. Takayama and Inoue [24] represented various experimental and computational results of the benchmark problem of planar shock wave diffraction over a 90° corner.

The present article describes the shock wave diffraction over various convex slabs (60°, 90°, and 120°) for 2 M_s (1.65, and 3.0). The isopycnics (contours of constant density) were shown in most of the previous studies. In the present study; isobars, isotherms, and the velocity contours over the slabs are plotted as well. Extensive analysis is performed on the basis of the pressure ratios and specific heat fluxes along the entire length of the step wall (for the diffraction at the first corner), which were previously evaluated at the wall shocks and at the last running expansion waves only.

We have observed that there isn’t enough study centered specifically on the vortexlet formation especially for different slab angles. In the present study, our main objective is to probe into flow physics with respect to the vortexlet formation implementing a finite volume-based high-order solver. Keeping that in mind, we studied the problem thoroughly for M_s = 1.65, and M_s = 3.0.

Methodology

Mathematical formulation

Flow separation is due to the adverse pressure gradient and expansion of flow area. Now, it is sudden change in area in the convex corners while it is continuous for round geometry. Thus, the inviscid modeling is applied here as it is also found suitable in several previous studies as already stated [2, 10, 11–12, 21, 22–23].

Conservation laws

The governing differential equations are assumed as Euler equations and the dependent variables are density, velocities in the x- and y- directions, pressure, and specific energy denoted by ρ, u, v, p, and e, respectively.

The set of 2-D conservative Euler equations is written as follows:

1

$$\frac{\partial Q}{\partial t}+\frac{\partial F}{\partial x}+\frac{\partial G}{\partial y}=0$$

Conserved variables vector Q, and vectors of inviscid fluxes F and G are given below.$$\begin{array}{cc}& Q=\left\{\left(\begin{array}{c}\rho \\ \rho u\\ \rho v\\ e\end{array}\right\}\right),\hfill \\ \hfill & F=\left\{\left(\begin{array}{c}\rho u\\ \rho u^2+p\\ \rho uv\\ \left(e+p\right)u\end{array}\right\}\right),\hfill \\ \hfill & G=\left\{\left(\begin{array}{c}\rho v\\ \rho vu\\ \rho v^2+p\\ \left(e+p\right)v\end{array}\right\}\right)\hfill \\ \hfill \end{array}$$

Pressure p is represented by the perfect gas equation in the following:

2

$$p=\left(\gamma -1\right)\left\{e-\frac{1}{2}\rho \left(u^2+v^2\right)\right\}$$

Integrating Eq. (1) over the computational domain Ω bounded by the surface ∂Ω and thereby using the theorem of Gauss divergence, the subsequent equation is established.

3

$$\frac{\partial }{\partial t}{\iint }_{\mathrm{\Omega }}Q\text{d}\mathrm{\Omega }+{\int }_{\partial \mathrm{\Omega }}\left(n_{x}F+n_{y}G\right)\text{d}S=0$$where $n_x$ and $n_y$ are the Cartesian components of the unit vector normally outwards to the differential surface element dS.

Above-mentioned equation can also be written as in the following on a Control Volume (CV) having ‘Nedge’ number of edges.

4

$$\frac{\text{d}}{\text{d}t}\left(Q_j{A}_j\right)+\sum _{m=1}^{\mathit{Nedge}}{F}_{edm}\mathrm{\Delta }{S}_m=0$$$A_j$ gives area of the CV numbered ‘j’. The local face number of CV is denoted by the subscript ‘m’. ‘Nedge’ represents the limiting edge numbers of CV, whereas ‘ΔS’ is the length of m.

Initial and boundary conditions

The region ahead of the moving shock is initialized by atmospheric (1 bar, 300 K) conditions. Pressure and density behind IS can be calculated by the following equations, representing p.r. (pressure ratio) and d.r. (density ratio).

5

$$\frac{p_2}{p_1}=1+\frac{2\gamma }{\gamma +1}\left(M_s^2-1\right)$$

6

$$\frac{{\rho }_2}{{\rho }_1}=\frac{\left(\gamma +1\right)M_s^2}{2+\left(\gamma -1\right)M_s^2}$$

The t.r. (temperature ratio) is evaluated using above-mentioned Eqs. (2, 5, and 6). Table 1 lists the p.r., d.r., and t.r. for all the M_s considered in present study. These values are necessary for patching between the high-pressure and low-pressure regions of the computational domain.

Table 1. Parameters in the present test cases

Ms	p.r	d.r	t.r	p2 (Pa)	ρ2 (kg/m3)	T2 (K)	up (m/s)
1.65	3.01	2.116	1.423	300,950	2.4561	426.9	302
3	10.33	3.857	2.679	1,033,000	4.47798	803.7	771.526

All the edges are considered as fixed walls. Velocities are zero at boundaries for no-slip and non-penetrating conditions.

Solver details

ANSYS Fluent solver was chosen to solve the governing differential equations. It had been validated previously also with both experimental and numerical results [2, 3]. The details of the setup of the solver is as follows:

• For spatial discretization, a third-order MUSCL (Monotonic Upstream-centered Scheme for Conservation Laws) scheme and Least Squares Cell-Based gradient were chosen. Whereas, for transient formulation, a second-order implicit scheme was selected.

• AUSM type flux splitting was adopted for the implicit solver.

• The convergence criterion was set as 1e−8.

• CFL (Courant–Friedrichs–Lewy) number = 0.5 here. It is the condition of convergence for numerical stability.

• Time-step size was taken in the order of 1e−7 s.

Heat flux equation

Heat flux is originated because of the temperature difference. Therefore, the corresponding values along the length of step-edge wall and at the wall shock were calculated.

7

$$q=\rho u{c}_p\mathrm{\Delta }T$$

However, we evaluated the specific heat flux by normalizing with the air molecular weight considering it to be an ideal gas.

Vorticity equation

8

$$\frac{D\omega }{\mathit{Dt}}=-\omega \left(\mathrm{\nabla }.V\right)+\frac{1}{{\rho }^2}\left(\mathrm{\nabla }\rho X\mathrm{\nabla }p\right)$$

To realize the vortexlet formation mechanism, we considered the Vorticity Transport Equation (VTE) given in Eq. (8). This form of equation neglects vortex tilting and stretching for velocity gradients (since 2-D flow is considered), vorticity diffusion due to viscosity, and body forces. Equation (8) represents the total vorticity (ω) variation in the field of flow.

• The first term on the R.H.S. denotes vortex stretching because of compressibility, VSC.

• The second term is vorticity production due to baroclinic effects, BAR.

The BAR produces vorticity because of misalignment of the gradients of pressure and density. That term includes the cross product of density and pressure gradients. So, if these gradients are not parallel, BAR is non-zero. The VSC affects vorticity magnitude because of either expansion or compression. Since, there is a negative sign in front of the VSC, an expansion results in a reduction in the vorticity magnitude. On the other hand, if the fluid is under compression, the vorticity is increased.

Computational domain

Figure 2 represents the computational domains for different convex (60°, 90°, and 120°) slabs. The variables p₂, ρ₂, T₂, and u_p represent static pressure, density, static temperature, and shock-induced gas velocity, respectively, in the region behind IS. Similarly, p₁, ρ₁, T₁, and u represent corresponding conditions ahead of IS, respectively; except for the fact that gas velocity is assumed as zero, i.e., stagnant conditions are prevailing there. Shock waves are marked in the designated positions at the commencement of the computation.

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Fig. 2

Problem setup for a 60°, b 90°, and c 120° slabs

Mesh refinement

For scheme validation purposes, mesh/grid independence test of the results were being carried out. Number of grid elements were varied from 196,257 to 812,022 and solutions were saved throughout the length of the step wall. We can observe from Table 2 that with the mesh becoming finer gradually, the density variation also diminishes. Densities at first step corner were extracted for mesh independence study on the 60° slab. Finally, a mesh size of 692,093 elements was chosen for optimum resolution.

Table 2. Mesh independence

Mesh elements	Density (kg/m3) difference
196,257	0.025
436,505
692,093	0.007
812,022	0.0005

Figure 3 is showing the domain of computation and view of meshes around the second corner of the 60° slab. Before the setup was built, mesh size was made finer ahead of the moving shock. The grid size is 0.028 cm in region ahead of IS and it is 0.047 cm in the back.

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Fig. 3

Computational domain of mesh

Results and discussions

The shock wave diffraction is self-similar due to the adiabatic and inviscid assumptions. This means the physical variables do not alter along the path of line integral enclosing the region behind UDSW. However, the path grows linearly with time for the unsteady flow field. The maximum M_s value was taken as 3.0. Beyond that, the IS, UDSW and LREW would also be reflected from the walls.

Validation

We had calculated the diffracted wall shock Mach number (Y co-ordinate values normalized by at) for the 90° step angle and 4 M_s values (1.65, 2.0, 2.5, and 3.0). Here, a stands for ambient sound speed and t represents the elapsed time since the IS passed the step-edge wall. Figure 4 shows the comparison of those values with the experimental observations by Skews [20] and the numerical results of Hillier [11]. As can be seen, our results show optimum agreements. The marginal deviation from the experimental results is due to the difference in initial conditions and the transition from subsonic to supersonic post-shock flow for intermediate M_s values.

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Fig. 4

Diffracted wall shock Mach number vs. M_s. Black square Present study, Red circle Hillier [11], Green triangle Skews [20] (color figure online)

Figure 5 indicates qualitative validation of numerical contours with experimental Schlieren images [9] over round geometry. In both the cases, the radius of the curve was taken as 2.8 mm. It also marks the flow development with time for unsteady cases. For 60 μs, contours are not distinctively visible for numerical and experimental results as well. However, in the case of 120 μs, some of the unique features are pointed out here. The InT (Internal Terminator) is formed because the flow becomes locally supersonic after being passed through the convergent–divergent passage formed by the solid wall, shear layer and the main vortex. This InT is further disintegrated into weak compression waves, like RS (Recompression Shock) and SInS (Secondary Internal Shock). Those waves have the main purpose of bridging the pressure gap between opposite sides of the shear layer.

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Fig. 5

Diffraction with M_s = 1.6 over a rounded corner a Schlieren sequence [9], b and c present numerical study

Vortexlet formation

Figure 6 is purposed to visualize almost all the flow features including the vortexlets in general by the numerical density gradient contour. Due to an expansion of the flow area, boundary layer is separated in the wall upstream. Then the slipstream is evolved as a separated shear layer which finally rolls back into PV as shown in Fig. 6. In the present study, vortexlet (VLP) is mainly observed around this primary vortex. ‘X’ and ‘Y’ in the subsequent Figs. represent the co-ordinate axes for the contours. At t = 0, the incident shock was situated (given in Figs. 1 and 2) at X = 0.2 m and Y = 0.15 to 0.3 m.

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Fig. 6

Density gradient contour over 90° slab for M_s = 1.65 at 200 μs

Figure 6 indicates the density gradient magnitudes over 90° slab when the flow simulation is up to 200 μs. The UDSW is generated from shock diffraction at first corner. Change in the gradient and curvature is observed when it becomes the incident shock for the second corner. The curved diffracted shock becomes horizontal, i.e., perpendicular or normal shock relative to the vertical wall of the slab. The REW (sometimes referred to as oblique shock) and secondary shock (here termed as SW in general) form an almost lambda structure. There is another such pattern observed before the VLP formation due to ES. The two ES are visibly prominent here and VLP is formed around the PV only before LDEW touches CS and/or PV. For the second corner, LDEW is generated from the point where the curvature of UDSW becomes changed. SV is also clearly visible, although it extends to a very small region.

The ratios of the minimum and maximum distances from the core of the PV to its extreme end are given in Table 3 for M_s = 1.65 to investigate the effects of corner angles on the flow field. This distance ratio is an important parameter which indicates how much a perfect circular vortex have been deformed. It also stands for the ratio of the minor to the major diameters of an elliptical structure. Table 3 clearly indicates that the vortex is more stretched or elliptical for lower corner angles. For the 60° corner, the wall effect and interaction of LDEW with the PV for larger duration create a major change in the orientation of the ideal vortex shape. It can be observed from subsequent figures, that the LDEW travels much further for 60° slab in comparison with other corners. For those reasons, the PV is elongated in the direction of flow and becomes more deviated from circular shape [see Fig. 7].

Table 3. Ratios of distances from the center of PV

60° slab	90° slab	120° slab
0.33	0.54	0.57

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Fig. 7

Contours over 60° slab of a shock, b density gradient for M_s = 1.65 at 320 μs

It was already observed that for M_s = 1.3, VLT, the first vortexlet is found behind the trailing edge of SL [12]. Whereas, for other M_s values, vortexlets (VLP) appear first around the PV and subsequently behind the trailing edge. For M_s = 1.65, VLP is formed right before LDEW reaches PV [see Fig. 6]. Like M_s = 1.65 case, vortexlets might have appeared around PV for M_s = 3.0. Although, it happens after the interaction of DS (Decelerated Shock) with PV [12] for M_s = 2.0. Since in our test case, DS and AS (Accelerated Shock) did not appear as such, the vortexlet formation for higher M_s values could not be captured. AS and DS are the products of interaction of lower diffracted Mach stem with PV and subsequent transformation of the Mach stem.

60° slab

Figure 7 depicts numerical simulation-based results for shock diffraction over adjacent 60° corners. M_s = 1.65 in Fig. 7a and b and there exists subsonic flow behind IS based on the induced flow velocity. It is evident already because the LREW travels back upstream from the first corner as shown in Fig. 7a. The circular expansion waves have mostly been reflected from upper wall. The starting point of the change in shock curvature is denoted by the meeting point of LDEW and UDSW, since portion of the shock wave above this do not have the knowledge of presence of the second corner. The region in the perturbed flow is also demarcated where influence of second corner is perceived. From both the corners, vortex is shed but the span is much more from the first corner because of longer exposure to flow and association with a higher oncoming velocity.

Figure 8 shows a case where M_s = 3.0 and flow behind IS is supersonic. LREW does not move back and from the corner. Steady Prandtl–Meyer expansion waves are being developed instead. Flow through this fan achieves a higher velocity than that behind the incident wave. That is how the SSW is generated which is bifurcated further to indicate its entry into the expansion regime from the shock-induced one. SV is distorted in shape because it is propagating into the non-uniform flow. The CS also appears to be distorted here as it interacts with the vortex and LDEW. Due to the second corner effects, shock curvature changes. SCS is also clearly visible which indicates that change. Exactly at the right side of where LDEW meets the UDSW, the shock does not possess any knowledge of another corner and appears to be perpendicular or normal to the 60° wall.

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Fig. 8

Density gradient contour for M_s = 3.0 at 180 μs over 60° slab

An increase in the M_s causes a rise in the expansion. This can be observed from Fig. 8, where the REW is visibly larger for M_s = 3.0 in comparison with the same for M_s = 1.65 [see Fig. 7a, b]. For greater M_s values, the flow velocity becomes supersonic around PV after passing through the expansion fan. Subsequently, strong SSW is formed behind REW [see Fig. 8]. The flow velocity becomes supersonic inside the PV. Thus, an embedded shock, ES, is generated for both the cases. Contours in Figs. 7, 8 are compared because in both the cases, the IS has traveled to the same location for different M_s and timespan of the simulations. The shock waves associated with the PV has traversed more distance for M_s = 3.0 as seen in Fig. 8, where the LREW does not travel back like in Fig. 7.

90° slab

Figures 9 and 10 represent shock diffraction over a 90° slab for about 320 μs. Unlike Fig. 6, here the LDEW reaches further and interacts with the CS, VLP, and PV. It is that instant, where VLT formation commences. The starting point of the shock curvature is denoted by the meeting point of LREW and incident normal shock wave. It is because portion of the shock wave above this do not have the knowledge of presence of any kind of corner. Similar phenomenon is observed for the second corner also. There is subsonic flow as the LDEW propagates upstream. As per authors’ knowledge, earlier studies mostly dealt with isopycnics. We have additionally shown the isobars, isotherms, and velocity contours as well [see Fig. 10]. The LREW traversed upstream of a shock for M_s ≤ 2.0. The pressure, temperature, and density get reduced along series of expansion waves or expansion fan due to rarefaction. The velocity is increased along expansion fan. This highlights the fact that the expansion wave is accelerating in nature. Vortex shocks are seen adjacent to the vortex core [see Figs. 9 and 10].

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Fig. 9

Density gradient contours over 90° slab for M_s = 1.65 at 320 μs

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Fig. 10

Contours over 90° slab for M_s = 1.65 at 320 μs

Figure 11 represents the density gradient contour over the benchmark problem geometry of 90° corners and/or slab for M_s = 3.0. It is noteworthy that the LDEW does not travel up to the same span as of Fig. 8 because of the geometrical features. That’s why the CS is not so distorted here like in the 60° slab. The disturbance front from the second corner (LDEW) travels visibly back to the flow below the SL, PV and the CS. The LDEW is in turn accelerating the ES at the left of the PV. The vertical position of the meeting point of the corresponding ES with the wall is clearly different from Fig. 8. Additionally the ES in the right side of the PV can be visually distinguished here, whereas, it was merged under the SSW in other slabs for M_s = 3.0.

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Fig. 11

Density gradient contour for M_s = 3.0 at 180 μs over 90° slab

Table 4 represents the strength of the embedded shocks as the pressure ratios across the shocks. Furthermore, with the increase in M_s, the PV is deformed more and the strength of ES also increases.

Table 4. Strengths of embedded shock for M_s = 1.65, and M_s = 3.0 over 90° slab at 180 μs

Ms	ES
1.65	1.67
3.0	2

120° slab

We can observe from Fig. 12 that for the 120° slab, only one shock triangle/lambda structure is being formed on SL comprised of SW and REW. Whereas, in other cases, two such patterns are visible. The SV is also not quite prominent here. The UDSW reaches further before meeting as wall shock but the LDEW does not get a chance to interact with the core of the vortex for the duration of 320 μs. Reflected expansion waves are visible, especially in Fig. 12b. Although, VLT is not seen here just like the 60° slab as it is a unique feature of the sharp-edged separation (90° corner).

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Fig. 12

Contours over 120° slab of a density gradient, and b shock for M_s = 1.65 at 320 μs

Figure 13 presents the density gradient contour over 120° slab for M_s = 3.0 at 180 μs. The shock curvature change of the UDSW happens at the very end of the simulation here as the LDEW travels to a barely minimum distance. As the PV is moved far away from the slab wall, the ES does not get to meet the wall here. The gradients of the REW, and the PV are also very much weakened.

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Fig. 13

Density gradient contour for M_s = 3.0 at 180 μs over 120° slab

Temporal characteristics of the unsteady waves

Figure 14 shows Shock contours for two different instants of time. At M_s = 1.65, first vortexlet VLP is generated under the influence of ES-generated instability before LDEW reaches the PV [see Fig. 14a]. However, at 320 μs, LDEW interacts with main vortex and VLP [see Fig. 14b]. Here, we can observe the initiation of VLT formation. Also, the LREW is distinctively visible at this instant. Whereas, at 200 μs, the location of the LREW is not prominent.

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Fig. 14

Shock contours over 90° slab for M_s = 1.65 at a 200 μs, b 320 μs

Downward movement of SL along slab wall

The downward shift of the SL/slipstream along the diffraction edge is clearly visible in the case of higher M_s (= 3.0 in present test cases) for 60° slab angle. So, the origin of the separation of flow is shifted from the first corner point. This indicates that for smaller step angles, flow separation is dependent on viscosity. It can be observed that for the second corner, shear layer is initiated from the corner point itself (since it is 120° relative to the flow). In Fig. 15, REW, bifurcated SSW, SL, PV, SV, distorted CS, SCS, and UDSW are all clearly perceived.

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Fig. 15

Vorticity (/s) contour over 60° slab for M_s = 3.0 at 180 μs

Pressure ratios and heat fluxes at 90° slab

Static pressures on the step-edge wall for the first diffraction had been taken from post processing. These were normalized using the atmospheric pressure ahead of IS. This newly generated parameter is termed ‘pressure ratio’. Extracting ρ, u, and T along the first diffraction wall and thereby utilizing Eq. (8), heat flux was evaluated. Figure 16a indicates that there exists a location where abrupt drop of pressure is observed implying the core of PV after a certain distance (around 0.027 m) along the step edge. Till that, expansion fan is dominant. Then there is sharp increase in the pressure ratio. UDSW and ES are the predominant factors here. Compressive nature through the shock wave is being reflected by such phenomenon. There exists two regions of sharp drops in the specific heat fluxes [see Fig. 16b]. At the very beginning, it is due to the expansion effects and the second one indicates the core of the vortices which is at the same location as in the pressure plot. However, there is a region (around 0.045 m) where the sharp increase in the heat flux value is hindered. This is an indicator of reattachment of the separated streamlines. The strong first running expansion wave just behind the UDSW near the slab wall is a major contributor in the flow separation. The temperature at the wall shock and gas temperature should be equal since there is zero heat flux at the point.

[See PDF for image]

Fig. 16

Variation of a Pressure Ratio and b Specific Heat Flux along the diffraction wall of the 90° slab

Conclusions

A numerical analysis of shock wave diffraction was bestowed over different convex slabs (60°, 90°, and 120°) and for 2 M_s values (1.65, and 3.0). Flow characteristics in perturbed region (secondary shock, shear layer, vortex, embedded shocks, vortexlets, and vortex sheet) were captured precisely. Along with isopycnics, isobars, isotherms, and velocity contours were also plotted.

The pressure ratio and heat flux were presented graphically along the step wall relative to the first diffraction. The downstream movement of the shear layer along the diffraction wall was observed in the case of higher Mach value for a slab angle of 60° in the current study.

The present study mainly focuses on vortexlets with the help of a high-order, compressible, and parallel solver. The trailing edge vortexlet is formed after the interaction between lower diffracted expansion wave and primary vortex; whereas, the embedded shock creates vortexlet around the primary vortex. There is significant difference between the location and time of occurrence of these two vortexlets. Our key observations are summarized below.

• The vortexlets appear around the PV and behind the trailing edge.

• For M_s = 1.65, most vortexlets are generated around PV by the ES originated inside core of PV.

• The maximum and minimum distances from the vortex core significantly affect the flow physics as well as the shape of the PV. As the span of travel of LDEW is maximum for 60° slab, vortex is also more stretched for the same.

• The distance traversed by the UDSW before meeting as wall shock is proportionately increased with the corner angles of the slabs.

Acknowledgements

The authors would like to thank the Departmental Head (Aerospace Engineering and Applied Mechanics, IIEST, Shibpur) for giving access to high performance computing facilities in a CAD laboratory, which is 24-h accessible.

Data availability

Data sets obtained in this research are available from the corresponding author through e-mail on reasonable request.

Declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose which are related to the work reported in this paper.

Abbreviations

2-D

Two dimensional

CS

Contact surface

d.r.

Density ratio

DTP

Double triple point

ES

Embedded shock

FDS

Flux difference splitting

FVS

Flux vector splitting

InT

Internal terminator

IS

Incident shock

LDEW

Lower diffracted expansion wave

LREW

Last running expansion wave

MUSCL

Monotonic upstream-centered scheme for conservation laws

p.r.

Pressure ratio

PV

Primary vortex

RS

Recompression shock

SInS

Secondary internal shock

SL

Shear layer

STP

Single triple point

SV

Secondary vortex

t.r.

Temperature ratio

UDSW

Upper diffracted shock wave

VTE

Vorticity transport equation

Subscripts

1

Parameters ahead of incident shock

2

Parameters behind incident shock

Greek symbols

γ

Specific heat ratio

ρ

Density

William Wolf

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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