Figure 5.8: Possible Kelvin-Helmholtz instability of the primary shear layer for 60 ° cone at MApex of 1.11
The extended test campaign considered all except the highest apex Mach Numbers possible on the apparatus (due to safety restrictions) and only tests in which the flow was well centred on the axis of the apparatus were considered (see § 5.1.4.2 for more on this distinction). The description of the basic flow field will be based on the images for the M Apex of 1.11 flow in the left of Figure 5.9. In frame a) the shear layer can be seen extending between the triple points (one marked as TP in the figure) of the Mach reflection and the vortex ring toward the centre of the frame. While there are other vortex rings shed **from** the outer **diffraction** lip at later times for most of the Mach numbers, the vortex ring of interest here is the first one caused by the axial jet formed when the **shock** **wave** first reflects at the cone apex. It should be noted that the shape seen here of the line connecting the tangent edges of the shear layer defines a concave cone where the outer edges of the cone are at the same level as the protruding portion closer to the centre line. The angle between the incident, reflected, and Mach stem waves is already quite shallow at this time. While some striations are visible in the shear layer near the centreline, the tangent edge of the shear layer still appears smooth. In frame b), there is noticeable striation of the tangent edge of the shear layer suggesting a KHI. The shear layer is still a concave cone but the entire cone now appears to extend away **from** the plane of the base defined by the outer edge and the curvature is noticeably lower than at earlier times.

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2.1 Geometric **Shock** Dynamics
In Whitham’s theory, knowledge of the shock’s geometry and Mach number along its profile are sufficient to predict how it will evolve. In his 1956 paper, Whitham [ 13 ] applied this method to two dimensional **shock** waves diffracting on convex surfaces and reflecting on concave surfaces. A distinction is made between **diffraction** and reflection wherein, **diffraction** is characterised by expansion waves and reflection by compression waves. In addition **shock** **diffraction** and reflection, Whitham also used GSD to investigate the stability of **shock** waves with **non**–uniform initial profiles (portions of the **shock** being planar while some portions are concave). In its formulation, an arbitrary **shock** shape was considered making the theory applicable not only to Whitham’s preliminary investigations, but to a multitude of other cases.

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Understanding of **shock** **wave** interaction with three-dimensional vortex loop is also a key issue for noise generation. Shimizu et al. [22] experimentally and theoretically investigated the mechanism of noise generation in **shock**-vortex ring interaction in a three-dimensional flow field. They focused on investigating noise generation at the early stage of the interaction. Noise generation comes **from** the scattered waves involving the **shock** **diffraction**, the acoustic **wave**, and the backward scattering by density inhomogeneity. Inoue et al. [32] numerically investigated sound generation in a long interaction process and showed large sound pressures occur due to **shock** **wave** focusing. **Shock** **wave** focusing in **shock**-vortex ring interaction had also been observed computationally by Takayama et al. [29]. Therefore, **shock** **wave** deformations such as **diffraction**, reflection, and focusing may lead to enhanced noise generation. Since **non**-circler vortex loops have self-induced vortex deformation, it leads to a more complicated mechanism for sound generation. This study focuses on investigating three-dimensional **shock** **wave** distortion and propagation phenomena. An experimental investigation of **shock**-square vortex loop interaction was conducted at an incident **shock** Mach number of 1.39 in a square cross-sectional open-end **shock** **wave** generating tube. The high-speed shadowgraph photography technique was used to evaluate flow characteristics.

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The study of heterogeneous and anisotropic materials and **non** normal impact geometries neces- sitated the development out of plane velocity measurements. These measurements were subject to the same bandwidth limitations as those faced by normal velocity interferometers. Just as VISAR provided a solution for normal motion, one of several designs implemented to desensitize transverse displacement interferometers for use with slow recording equipment is the Variable Sensitivity Dis- placement Interferometer (VSDI) developed at Brown University [35]. VSDI is a **diffraction** assisted displacement interferometer capable of resolving both normal and transverse motion of a grating affixed to the rear surface of a target plate in gas gun experiments. Its primary use is to determine the structure and magnitude of a shear **wave** pulse traveling through a solid material undergoing combined pressure and shear loading. This experimental arrangement, known as Pressure Shear Plate Impact (PSPI) generates combined loading in a thin foil of a material of interest sandwiched between two high **shock** impedance anvils. The surface normal of the composite stack is then inclined relative to the direction of impact. A keyed gas gun is then used to propel a similarly inclined flyer plate, which generates normal and transverse motion in the target plate due to the impact direction being inclined relative to the impact surfaces.

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The speed of the arterial pulse **wave** is commonly termed the pulse-**wave** velocity (PWV). It has been shown to increase during the course of certain diseases, and this increase has generally been attributed to "arterial stiff- ness" [4-7]. This interpretation is consistent with Equa- tion (1), the Korteweg-Moens Equation, which is based in part on the assumption that the fluid is not viscous. Lambossy [8] introduced a model for arterial blood flow in which viscosity results in shear force on the inner wall of an artery and the pressure gradient is a simple har- monic function of time, e iωt . In this model, is a con- stant, and ω is the frequency of oscillation. Other constants in the model are he viscosity of blood, denoted μ , and the density, denoted ρ . The arterial wall is a straight, rigid cylindrical tube of radius R.

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In fact, the surface elasticity theory can be applied not only to the statics analysis, but also to the dynamics analysis. In the framework of surface elasticity theory, the scattering of plane compressional and shear waves by a single na- no-sized coated fiber and the multiple scattering by two cylinder inclusions, which embedded in an elastic matrix is studied by Ou and Lee using the method of eigenfunction expansion [7] [8]. Zhang et al . [9] used the **wave** function ex- pansion method to study the effect of nano-sized arrays on the longitudinal **wave** **diffraction** in elastic media, and gave the corresponding elastic **diffraction** fields. Ru et al . [10] considered the SV **wave** multiple scattering caused by a cluster of nano-cylindrical holes. Using the displacement potential method and the **wave** function expansion method, they derived the scattering field around the hole. Ou et al . [11] studied the effects of semi-cylindrical inclusions on the scattering of plane P **wave** in an elastic half-plane. The results show that surface energy has a significant effect on the scattering of plane P **wave** as the radius of the semi-cylindrical inclusions shrink to nanometers. Wang et al . [12] [13] used Gurtin’s surface elasticity model to analyze the **diffraction** of elastic **wave** by a nanosized inhomogeneity, and they also demonstrated that the surface energy has a significant effect on the **diffraction** of the elastic **wave** when the cavity ra- dius is reduced to the nanoscale.

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The evaluation of the surface integrals needs high computation times for **diffraction** problems with complex geometries. TBDW method reduces the surface integral to a line integral, resulting in a significant improvement in the computation time. Moreover, the line integral reduction of the surface integrals enables one to evaluate the edge diffracted fields directly, by integrating the reduced integrand along the edge contour.

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This study more deeply investigates the properties of such a new material which satisﬁes the fractional boundary condition. The theoretical description of electromagnetic plane **wave** scattering by the two strips with diﬀerent dimensions is given in the theoretical section. After that, the total electric ﬁeld distribution and Total Radar Cross Section (TRCS) are investigated and obtain numerical results. In the general case, the solution of the problem is reduced to a solution of a system of linear algebraic equations (SLAE) which can be obtained with a given accuracy. However, for the values of the fractional order (FO) α = 0 . 5, the solution is obtained in an analytical form which is the advantage

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+ with the integer n depending on the radius of cylinder. Generally, the integer n increases for a cylinder of larger diameter. The usual approximation by Laguerre functions is extended by introducing a scale parameter. The convergence of Laguerre series is then dependent on the value of the scale parameter s. The ana- lytical and numerical computations of series coefficients are performed to study the number of se- ries terms to keep the same accuracy. Indeed, the choice of integer n depends on the scale para- meter. Furthermore, **diffraction** waves generated by a semi-sphere inside the cylinder are eva- luated on the cylinder surface. It is shown that the approximation by Laguerre series for diffrac- tion waves on the cylinder is effective. This work provides important information for the choice of the radius of control surface in the domain decomposition method for solving hydrodynamic problems of body-**wave** interaction.

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In this paper we present a new approach for the study of Burgers equations. Our purpose is to describe the asymptotic behavior of the solution in the cauchy problem for viscid equation with small parametr ε and to discuss in particular the case of periodic **wave** **shock**. We show that the solution of this problem approaches the **shock** type solution for the cauchy problem of the inviscid burgers equation. The results are formulated in classical mathematics and proved with infinitesimal techniques of **non** standard analysis.

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In 90 of the 101 Achilles tendons (89%) with chronic painful mid-portion Achilles tendinosis, treatment was satisfactory and the patients were back on their pre-injury activity level after the 12-week training regimen. In these patients, the amount of pain during activity, registered on the VAS-scale decreased significantly **from** 6.7 to 1.0.

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Diffusers slow down the air entering the engines of supersonic aircraft to subsonic speeds to avoid damaging the engine. They typically do this by inducing **shock** waves prior to the engine inlet. In this report several diffuser geometries were modeled to cr eate oblique **shock** waves to reduce air speed with less stagnation pressure losses and drag than normal **shock** waves would create. These geometries include single ramp, double ramp, curved ramp, and a double ramp cone with external ramp and channel. This was done in Ansys Fluent using a refined mesh with an inflation layer along the diffuser surface. The CFD was run using a density based solver coupled with a turbulent model and the resulting stagnation pressure losses, drag, and boundary layer separation were compared.

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CHARACTERIZATION OF THE **SHOCK** **WAVE** STRUCTURE IN WATER
Emilie Maria Teitz, B.S. Marquette University, 2017
The scientific community is interested in furthering the understanding of **shock** **wave** structures in water, given its implications in a wide range of applications; **from** researching how **shock** waves penetrate unwanted body tissues to studying how humans respond to blast waves. **Shock** **wave** research on water has existed for over five decades. Previous studies have investigated the **shock** response of water at pressures ranging **from** 1 to 70 GPa using flyer plate experiments. This report differs **from** previously published experiments in that the water was loaded to **shock** pressures ranging **from** 0.36 to 0.70 GPa. The experiment also utilized tap water rather than distilled water as the test sample.

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computational burden indicators of SNNOLS and NNOLS are shown in Fig. 2. It is noticeable that the number of iterations L required to reach a support of cardinality K is larger than K because of support compression. Specifically, the histograms of Fig. 2(b) show that on average, L is larger than K for NNOLS and even larger for SNNOLS (the average values are given in the last columns of Tab. I). This is consistent with the fact that the NNOLS selection rule is more involved but more reliable. Moreover, the standard deviation of L corresponding to the histograms of Fig. 2(b) is 17 and 12 for SNNOLS and NNOLS, respectively, which indicates that the size of the support found after k iterations may significantly vary between trials. In order to get meaningful evaluations, we choose to compute the average values of each indicator over the last t iterations, with t ∈ {0, . . . , K − 1}. When t = 0, only the last iteration is taken into account, so the current support is of size K. For larger values of t, the supports found during the last t iterations have varying sizes but the averaging operation remains meaningful, especially for the last iterations, which are the most costly. The curves displaying the average of each indicator over 200 trials and over the last t iterations are shown in Fig. 2(a). One can observe **from** the curve (1 − ρ ↓ ) that the rate of **non**-descending atoms gradually

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Recently, Defina, Susin & Viero (2008) presented high-resolution numerical solutions of the depth averaged inviscid shallow water equations which provided new information on the weak **shock** reflection domain within the von Neumann paradox conditions. The authors computed **shock** reflections close to the Guderley and the Vasilev reflections which confirmed the validity of the four-**wave** theory, however they did not discover a complex sequence of supersonic patches predicted by Tesdall et al. (2002). The absence of the additional triple points and supersonic patches agrees with the suggestion by Vasil’ev et al. (2008) that the complex sequence of triple points only occurs during unsteady flow conditions, which is not the present case in the work of Defina, Susin & Viero (2008). It was noted that the four-**wave** model correctly predicts the **wave** pattern around the triple point but is not the solution of the GR, as the flow downstream of the Mach stem in the vicinity of the triple point is still supersonic and it is further turned towards the Mach stem. Defina, Susin & Viero (2008) therefore discuss a possible solution to better describe the developed **wave** characteristics of the GR. Note all results are based on the Froude number F 0 = 1.7

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to Illodel the interaction of a lithotripter shoek wave with two stable spherical bubble, and to observe: • the re'flection, transmission and refraction of the shock waves • the collapse[r]

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were generated for use on parallel processor systems. To reduce computational time on a large 3-D grid, inviscid wall boundaries were used for all nozzle surfaces. External flow conditions were run at Mach 2.2, an angle of attack of zero, and an Euler solution was generated.
The wedge (Figure 3) was unswept with 2.5° half angle leading and trailing edges. This wedge permitted study of **shock** and expansion regions passing through a nozzle plume. The wedge was located at 57.1 in. above the nozzle centerline, and the leading edge of the wedge was located in a plane 13.14 in. upstream of the nozzle exit. In this case, the axial station for the leading edge of the wedge was close to the nozzle throat, not the nozzle exit. The upper boundary of the mesh contained the profile of the lower wedge surface where the boundary conditions were inviscid wall boundaries. All other boundary conditions were again set to the WIND-US ‘freestream’ boundary condition.

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IX 3 Experimental Systems for Shock Compression of Solids C-l 3.1 High Velocity Planar Impact Loading System C-l 3.2 Experimental Techniques: Diagnostic Systems C-3 3.2.l Arrival Time De[r]

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Theory shows that the **shock** **wave** angle and the **shock** structure depend on the cone angle of the wind tunnel model, and on the Mach number and the def- lection angle of the incoming flow [1]. In the present study, the cone angle of the wind tunnel model and the Mach number of the incoming flow are fixed, an on-board plasma deflector is introduced to study **shock** structure modification by the flow deflection [20]. The polarity of the applied voltage enables electron plasma to be accelerated in the upstream direction by the applied electric field, it forms a plasma deflector in the upstream region to deflect incoming flow through elastic collisions. Ion plasma also affects the incoming flow, but via a different process. Ions moving through their own gas are subject to charge transfer to the neutral gas, which is a predominant inelastic collision process in the low ion energy regime [21]. An ion becomes a neutral particle after the charge transfer, but retains its velocity, which is usually low. Thus, these par- ticles do not contribute to the **shock** **wave** formation. The ions converted **from** neutrals through charge-transfer are collected by the cathode and do not con- tribute to the **shock** **wave** generation either. The **shock** of the deflected flow is expected to have a larger **shock** angle (than that of the baseline one) and a mod- ified structure, representing a weaker **shock**.

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