Mechanical and Aerospace Engineering Faculty Research & Creative WorksCopyright (c) 2020 Missouri University of Science and Technology All rights reserved.
https://scholarsmine.mst.edu/mec_aereng_facwork
Recent documents in Mechanical and Aerospace Engineering Faculty Research & Creative Worksen-usThu, 19 Nov 2020 08:12:22 PST3600Compliant Mechanisms -- Memory Lane and Some Novel and Exciting Applications
https://scholarsmine.mst.edu/mec_aereng_facwork/4525
https://scholarsmine.mst.edu/mec_aereng_facwork/4525Wed, 07 Oct 2020 10:10:54 PDT
A set of chance happenings led to the beginning of formalization of a research area, termed “compliant mechanisms,” around early 1980s. This presentation takes one down that memory lane describing how it all came about. What appeared to be a modest beginning, in time would exceed every expectation and more, and continue to proliferate with applications into multiple disciplines. Some very recent theoretical developments in compliant mechanisms are discussed, that offer fundamental discovery, with the potential for innovating exciting design applications.
]]>
Ashok MidhaSuppression of the Spherically Converging Magnetohydrodynamic Richtmyer-Meshkov Instablity in an Octahedrally Symmetric Seed Magnetic Field
https://scholarsmine.mst.edu/mec_aereng_facwork/4524
https://scholarsmine.mst.edu/mec_aereng_facwork/4524Thu, 20 Aug 2020 11:02:28 PDT
We present results of ideal magnetohydrodynamics simulations investigating the Richtmyer-Meshkov instability in near-spherical implosions in the presence of an octahedrally symmetric seed magnetic field. The problem is motivated by the desire to maintain a symmetrical collapse of the primary shock wave, minimally distorted by the effect of the seed magnetic field, while retaining the seed-field-induced suppression of the Richtmyer-Meshkov instability. The field is generated by a set of six current loops arranged around the target as on the faces of a cube. The instability is generated on a perturbed spherical density interface that is accelerated from the outside by imploding magnetohydrodynamic shocks, which are in turn generated by a spherical Riemann problem. The perturbation on the density interface is formed with a single-dominant-mode spherical harmonics expansion. We investigate the evolution of the interface and the transport of baroclinic vorticity near the interface, and examine the extent of the distortion to the primary magnetohydrodynamic shock system induced by the seed field.
]]>
Wouter Mostert et al.Effects of Magnetic Fields on Magnetohydrodynamic Cylindrical and Spherical Richtmyer-Meshkov Instability
https://scholarsmine.mst.edu/mec_aereng_facwork/4523
https://scholarsmine.mst.edu/mec_aereng_facwork/4523Thu, 20 Aug 2020 11:02:24 PDT
The effects of seed magnetic fields on the Richtmyer-Meshkov instability driven by converging cylindrical and spherical implosions in ideal magnetohydrodynamics are investigated. Two different seed field configurations at various strengths are applied over a cylindrical or spherical density interface which has a single-dominant-mode perturbation. The shocks that excite the instability are generated with appropriate Riemann problems in a numerical formulation and the effect of the seed field on the growth rate and symmetry of the perturbations on the density interface is examined. We find reduced perturbation growth for both field configurations and all tested strengths. The extent of growth suppression increases with seed field strength but varies with the angle of the field to interface. The seed field configuration does not significantly affect extent of suppression of the instability, allowing it to be chosen to minimize its effect on implosion distortion. However, stronger seed fields are required in three dimensions to suppress the instability effectively.
]]>
Wouter Mostert et al.Converging Cylindrical Magnetohydrodynamic Shock Collapse Onto a Power-Law-Varying Line Current
https://scholarsmine.mst.edu/mec_aereng_facwork/4522
https://scholarsmine.mst.edu/mec_aereng_facwork/4522Thu, 20 Aug 2020 11:02:20 PDT
We investigate the convergence behaviour of a cylindrical, fast magnetohydrodynamic (MHD) shock wave in a neutrally ionized gas collapsing onto an axial line current that generates a power law in time, azimuthal magnetic field. The analysis is done within the framework of a modified version of ideal MHD for an inviscid, non-dissipative, neutrally ionized compressible gas. The time variation of the magnetic field is tuned such that it approaches zero at the instant that the shock reaches the axis. This configuration is motivated by the desire to produce a finite magnetic field at finite shock radius but a singular gas pressure and temperature at the instant of shock impact. Our main focus is on the variation with shock radius r , as r→0 , of the shock Mach number M(r) and pressure behind the shock p(r) as a function of the magnetic field power-law exponent μ ⩾ 0 , where μ = 0 gives a constant-in-time line current. The flow problem is first formulated using an extension of geometrical shock dynamics (GSD) into the time domain to take account of the time-varying conditions ahead of the converging shock, coupled with appropriate shock-jump conditions for a fast, symmetric MHD shock. This provides a pair of ordinary differential equations describing both M(r) and the time evolution on the shock, as a function of r , constrained by a collapse condition required to achieve tuned shock convergence. Asymptotic, analytical results for M(r) and p(r) are obtained over a range of μ for general γ , and for both small and large r . In addition, numerical solutions of the GSD equations are performed over a large range of r , for selected parameters using γ = 5/3 . The accuracy of the GSD model is verified for some cases using direct numerical solution of the full, radially symmetric MHD equations using a shock-capturing method. For the GSD solutions, it is found that the physical character of the shock convergence to the axis is a strong function of μ . For 0 ⩽ μ < 4/13 , M and p both approach unity at shock impact r=0 owing to the dominance of the strong magnetic field over the amplifying effects of geometrical convergence. When μ ⩾ 0.816 (for γ = 5/3 ), geometrical convergence is dominant and the shock behaves similarly to a converging gas dynamic shock with singular M(r) and p(r) , r→0 . For 4/13 < μ ⩽ 0.816 three distinct regions of M(r) variation are identified. For each of these p(r) is singular at the axis.
]]>
Wouter Mostert et al.Local Field Effects on Magnetic Suppression of the Converging Richtmyer-Meshkov Instability
https://scholarsmine.mst.edu/mec_aereng_facwork/4521
https://scholarsmine.mst.edu/mec_aereng_facwork/4521Thu, 20 Aug 2020 11:02:17 PDT
We examine how the suppression of the converging shockdriven Richtmyer-Meshkov instability by an applied magnetic field is dependent on the local magnetic field strength and orientation. In particular, we examine whether the extent of suppression can be reasonably predicted by a linear model for the planar case. This is done for cylindrically converging cases with a high perturbation wavenumber and two different initial magnetic field configurations.
]]>
Vincent Wheatley et al.Geometrical Shock Dynamics for Magnetohydrodynamic Fast Shocks
https://scholarsmine.mst.edu/mec_aereng_facwork/4520
https://scholarsmine.mst.edu/mec_aereng_facwork/4520Thu, 20 Aug 2020 11:02:13 PDT
We describe a formulation of two-dimensional geometrical shock dynamics (GSD) suitable for ideal magnetohydrodynamic (MHD) fast shocks under magnetic fields of general strength and orientation. The resulting area–Mach-number–shock-angle relation is then incorporated into a numerical method using pseudospectral differentiation. The MHD-GSD model is verified by comparison with results from nonlinear finite-volume solution of the complete ideal MHD equations applied to a shock implosion flow in the presence of an oblique and spatially varying magnetic field ahead of the shock. Results from application of the MHD-GSD equations to the stability of fast MHD shocks in two dimensions are presented. It is shown that the time to formation of triple points for both perturbed MHD and gas-dynamic shocks increases as ϵ^(-1), where ϵ is a measure of the initial Mach-number perturbation. Symmetry breaking in the MHD case is demonstrated. In cylindrical converging geometry, in the presence of an azimuthal field produced by a line current, the MHD shock behaves in the mean as in Pullin et al. (Phys. Fluids, vol. 26, 2014, 097103), but suffers a greater relative pressure fluctuation along the shock than the gas-dynamic shock.
]]>
Wouter Mostert et al.Magnetohydrodynamic Implosion Symmetry and Suppression of Richtmyer-Meshkov Instability in an Octahedrally Symmetric Field
https://scholarsmine.mst.edu/mec_aereng_facwork/4519
https://scholarsmine.mst.edu/mec_aereng_facwork/4519Thu, 20 Aug 2020 11:02:10 PDT
We present numerical simulations of ideal magnetohydrodynamics showing suppression of the Richtmyer-Meshkov instability in spherical implosions in the presence of an octahedrally symmetric magnetic field. This field configuration is of interest owing to its high degree of spherical symmetry in comparison with previously considered dihedrally symmetric fields. The simulations indicate that the octahedral field suppresses the instability comparably to the other previously considered candidate fields for light-heavy interface accelerations while retaining a highly symmetric underlying flow even at high field strengths. With this field, there is a reduction in the root-mean-square perturbation amplitude of up to approximately 50% at representative time under the strongest field tested while maintaining a homogeneous suppression pattern compared to the other candidate fields.
]]>
Wouter Mostert et al.Singularity Formation in the Geometry of Perturbed Shocks of General Mach Number
https://scholarsmine.mst.edu/mec_aereng_facwork/4518
https://scholarsmine.mst.edu/mec_aereng_facwork/4518Thu, 20 Aug 2020 11:02:06 PDT
While planar shock waves are known to be stable to small perturbations in the sense that the perturbation amplitude decays over time, it has also been suggested that plane propagating shocks can develop singularities in some derivative of their geometry (Whitham (1974) Linear and nonlinear waves. Wiley, New York) in a nonlinear, wave reinforcement process. We present a spectral-based analysis of the equations of geometrical shock dynamics that predicts the time to singularity formation in the profile of an initially perturbed planar shock for general shock Mach number. We find that following an initially sinusoidal perturbation, the shock shape remains analytic only up to a finite, critical time that is a monotonically decreasing function of the initial perturbation amplitude. At the critical time, the shock profile ceases to be analytic, corresponding physically to the incipient formation of a “shock-shock” or triple point. We present results for gas-dynamic shocks and discuss the potential for extension to shock dynamics of fast MHD shocks.
]]>
Wouter Mostert et al.Spontaneous Shock-Shock and Singularity Formation on Perturbed Planar Shock Waves
https://scholarsmine.mst.edu/mec_aereng_facwork/4517
https://scholarsmine.mst.edu/mec_aereng_facwork/4517Thu, 20 Aug 2020 11:02:03 PDT
We discuss the evolution of perturbed planar gas-dynamic and magnetohydrodynamics shock waves. An asymptotic closed form solution of the equations of geometrical shock dynamics (GSD) based on spectral analysis is described that predicts a time to loss of analyticity in the profile of a plane propagating shock wave subject to a smooth, spatially-periodic shape and Mach number perturbation of arbitrarily small magnitude. The shock shape remains analytic only up to a finite, critical time that is found to be inversely proportional to a measure of the initial perturbation amplitude. It is also shown that this analysis can also be applied to strong, fast MHD shocks in the presence of an external magnetic field whose field lines are parallel to the unperturbed shock. The relation between this critical time and the numerical detection of the time to formation of shock-shocks (Mostert et al., JFM. 2017) will be discussed.
]]>
Wouter Mostert et al.Singularity Formation on Perturbed Planar Shock Waves
https://scholarsmine.mst.edu/mec_aereng_facwork/4516
https://scholarsmine.mst.edu/mec_aereng_facwork/4516Thu, 20 Aug 2020 11:01:59 PDT
We present an analysis that predicts the time to development of a singularity in the shape profile of a spatially periodic perturbed, planar shock wave for ideal gas dynamics. Beginning with a formulation in complex coordinates of Whitham’s approximate model geometrical shock dynamics (GSD), we apply a spectral treatment to derive the asymptotic form for the leading-order behaviour of the shock Fourier coefficients for large mode numbers and time. This is shown to determine a critical time at which the coefficients begin to decay, with respect to mode number, at an algebraic rate with an exponent of −5/2 , indicating loss of analyticity and the formation of a singularity in the shock geometry. The critical time is found to be inversely proportional to a representative measure of perturbation amplitude 𝜖 with an explicit analytic form for the constant of proportionality in terms of gas and shock parameters. To leading order, the time to singularity formation is dependent only on the first Fourier mode. Comparison with results of numerical solutions to the full GSD equations shows that the predicted critical time somewhat underestimates the time for shock–shock (triple-point) formation, where the latter is obtained by post-processing the numerical GSD results using an edge-detection algorithm. Aspects of the analysis suggest that the appearance of loss of analyticity in the shock surface may be a precursor to the first appearance of shock–shocks, which may account for part of the discrepancy. The frequency of oscillation of the shock perturbation is accurately predicted. In addition, the analysis is extended to the evolution of a perturbed planar, fast magnetohydrodynamic shock for the case when the external magnetic field is aligned parallel to the unperturbed shock. It is found that, for a strong shock, the presence of the magnetic field produces only a higher-order correction to the GSD equations with the result that the time to loss of analyticity is the same as for the gas-dynamic flow. Limitations and improvements for the analysis are discussed, as are comparisons with the analogous appearance of singularity formation in vortex-sheet evolution in an incompressible inviscid fluid.
]]>
Wouter Mostert et al.Spontaneous Singularity Formation in Converging Cylindrical Shock Waves
https://scholarsmine.mst.edu/mec_aereng_facwork/4515
https://scholarsmine.mst.edu/mec_aereng_facwork/4515Thu, 20 Aug 2020 11:01:56 PDT
We develop a nonlinear, Fourier-based analysis of the evolution of a perturbed, converging cylindrical strong shock using the approximate method of geometrical shock dynamics (GSD). This predicts that a singularity in the shock-shape geometry, corresponding to a change in Fourier-coefficient decay from exponential to algebraic, is guaranteed to form prior to the time of shock impact at the origin, for arbitrarily small, finite initial perturbation amplitude. Specifically for an azimuthally periodic Mach-number perturbation on an initially circular shock with integer mode number q and amplitude proportional to ε ≪ 1, a singularity in the shock geometry forms at a mean shock radius R_{u,c} ∼(q^{2}ε)^{-1/ b1} where b_{1} (γ) < 0 is a derived constant and γ the ratio of specific heats. This requires q^{2}ε ≪ 1, q ≫ 1. The constant of proportionality is obtained as a function of γ and is independent of the initial shock Mach number M_{0}. Singularity formation corresponds to the transition from a smooth perturbation to a faceted polygonal form. Results are qualitatively verified by a numerical GSD comparison.
]]>
Wouter Mostert et al.Energy Dissipation in Shallow Water Breaking Waves
https://scholarsmine.mst.edu/mec_aereng_facwork/4514
https://scholarsmine.mst.edu/mec_aereng_facwork/4514Thu, 20 Aug 2020 11:01:52 PDT
We present numerical results of energy dissipation in two-dimensional shallow water breaking waves. Using a two-phase DNS approach, we seek a fundamental energy dissipation model and classification scheme for breaking type and mechanism. A solitary wave of amplitude a 0 is initialized over a region of uniform depth h and propagated onto a beach with a uniformly sloping bathymetry of gradient α. We discuss the various types of resulting breakers as a function of these parameters, including plunging, spilling, and surging types. The breaker dissipates kinetic and gravitational potential energy in the wave before it runs up onto the beach. We discuss the energy dissipation and wave run-up in terms of the control parameters and propose a model for energy dissipation adapted from the inertial scaling model for deep water breakers.
]]>
Wouter Mostert et al.Bubble Pinch Off in a Turbulent Flow
https://scholarsmine.mst.edu/mec_aereng_facwork/4513
https://scholarsmine.mst.edu/mec_aereng_facwork/4513Thu, 20 Aug 2020 11:01:49 PDT
We present experiments and direct numerical simulations of the break-up of bubbles in a turbulent flow. The turbulent flow induces bubble deformation, creating a neck that then pinches off giving birth to child bubbles. We focus on the transition from the deformation regime controlled by the turbulent fluctuations to the final universal pinch-off, which present a self-similar dynamics, similar to the one reported in bubble pinch-off in still water. We characterize the large scale deformation of the bubble by its aspect ratio and observe fluctuations at the order of the eddy turnover scale, while the final pinch-off starts above a critical deformation, typically one millisecond prior to break-up. Over this last millisecond, the necking follows a universal self-similar behavior, independent of the turbulent flow.
]]>
Luc Deike et al.High-Resolution DNS of Breaking Waves
https://scholarsmine.mst.edu/mec_aereng_facwork/4512
https://scholarsmine.mst.edu/mec_aereng_facwork/4512Thu, 20 Aug 2020 11:01:45 PDT
We present bubble and droplet size distributions resulting from breaking ocean waves in deep water, using high-resolution three-dimensional direct numerical simulation. We use the open-source Basilisk code to simulate the viscous Navier-Stokes equations in two phases with surface tension at effective resolutions of up to 4096^{3}. The interface is represented and advected with a momentum-conservative volume-of-fluid scheme. The high effective resolutions are made possible with an octree adaptive mesh refinement scheme which is robustly implemented in Basilisk. The wave is initialized in one wavelength with an unstable third-order Stokes formulation, which produces local conditions leading to a plunging breaker which entrains air and ejects spray, which are directly resolved by the mesh. Varying the Bond and Reynolds numbers, which control surface tension and viscosity relative to the gravitational and inertial effects respectively, we discuss issues such as bubble breakup in turbulent flow; dimensionality in the transition to turbulence; droplet production and breakup; and numerical grid convergence.
]]>
Wouter Mostert et al.Direct Numerical Simulations of Bubble Break-Up in Turbulence
https://scholarsmine.mst.edu/mec_aereng_facwork/4511
https://scholarsmine.mst.edu/mec_aereng_facwork/4511Thu, 20 Aug 2020 11:01:41 PDT
We present direct numerical simulations of bubble break-up in a three-dimensional homogeneous and isotropic turbulent flow. We consider the effect of the turbulent Reynolds number and the bubble Weber number on the break-up dynamics, the number of child bubble created together with their size and the break-up frequency. An ensemble of simulation is done in order to study these quantities statistically. For Weber number slightly above the critical value number, we retrieve binary break-up with two child bubble of similar size, while for large Weber number, we observe more complex break-up patterns with successive break-up events and the formation of a large number of much smaller bubbles.
]]>
Luc Deike et al.Direct Numerical Simulation of Rain Drop Impact on a Thin Layer of Oil over a Deep Water Pool
https://scholarsmine.mst.edu/mec_aereng_facwork/4510
https://scholarsmine.mst.edu/mec_aereng_facwork/4510Thu, 20 Aug 2020 11:01:38 PDT
The impact of a water droplet onto a deep pool coated by a film of oil has not yet been thoroughly investigated numerically in the large Weber number range. This process occurs during rainfall on oil slicks at sea, and ejects oily aerosols into the atmosphere that later forms atmospheric particulates. We present direct numerical simulations of the three-phase process using the solver Basilisk. The numerical results are qualitatively and quantitatively compared to existing experimental data, and discuss the influence of numerical resolution on the crown and canopy closure. Finally, the effects of the oil properties and drop shape upon impact on the resulting splash dynamics are investigated.
]]>
Francis Ogoke et al.Bubble Pinch-Off in Turbulence: Shape Oscillations and Escaping Self-Similarity
https://scholarsmine.mst.edu/mec_aereng_facwork/4509
https://scholarsmine.mst.edu/mec_aereng_facwork/4509Thu, 20 Aug 2020 11:01:34 PDT
Though bubble pinch-off is an archetype of a dynamical system evolving towards a singularity, it has always been described in idealized theoretical and experimental conditions. Using experiments, simulations, and analytical modeling, we consider bubble pinch-off in a turbulent flow, representative of natural conditions in the presence of strong and random perturbations. We show that the turbulence sets the initial conditions for pinch-off, but once the pinch-off starts, the turbulent time at the neck scale becomes much slower than the pinching dynamics: the turbulence freezes. We show that the average neck size, d , can be described by d (t -t_{0})α , where t_{0} is the pinch-off, or singularity time, and α 0 . 5 , in close agreement with the axisymmetric theory with zero initial flow. Neck shape oscillations set by the initial conditions are described by a quasi-two-dimensional linear perturbation model, and persistent asymmetries in the neck are related to the complex flow field induced by the deformed bubble shape. In many cases, a three-dimensional kink-like structure forms on part of the neck just before pinch-off, causing d to escape its self-similar decrease.
]]>
Daniel J. Ruth et al.Bubble Pinch-Off in Turbulence
https://scholarsmine.mst.edu/mec_aereng_facwork/4508
https://scholarsmine.mst.edu/mec_aereng_facwork/4508Thu, 20 Aug 2020 11:01:31 PDT
Although bubble pinch-off is an archetype of a dynamical system evolving toward a singularity, it has always been described in idealized theoretical and experimental conditions. Here, we consider bubble pinch-off in a turbulent flow representative of natural conditions in the presence of strong and random perturbations, combining laboratory experiments, numerical simulations, and theoretical modeling. We show that the turbulence sets the initial conditions for pinch-off, namely the initial bubble shape and flow field, but after the pinch-off starts, the turbulent time at the neck scale becomes much slower than the pinching dynamics: The turbulence freezes. We show that the average neck size, d̅, can be described by d̅ ~ (t − t_{0})^{α}, where t_{0} is the pinch-off or singularity time and α ≈ 0.5, in close agreement with the axisymmetric theory with no initial flow. While frozen, the turbulence can influence the pinch-off through the initial conditions. Neck shape oscillations described by a quasi-2-dimensional (quasi-2D) linear perturbation model are observed as are persistent eccentricities of the neck, which are related to the complex flow field induced by the deformed bubble shape. When turbulent stresses are less able to be counteracted by surface tension, a 3-dimensional (3D) kink-like structure develops in the neck, causing d̅ to escape its self-similar decrease. We identify the geometric controlling parameter that governs the appearance of these kink-like interfacial structures, which drive the collapse out of the self-similar route, governing both the likelihood of escaping the self-similar process and the time and length scale at which it occurs.
]]>
Daniel J. Ruth et al.Bubble and Droplet Size Distributions in Breaking Waves
https://scholarsmine.mst.edu/mec_aereng_facwork/4507
https://scholarsmine.mst.edu/mec_aereng_facwork/4507Thu, 20 Aug 2020 11:01:27 PDT
We present size distributions of bubble and droplet populations generated by breaking waves using high-resolution direct numerical simulation of the two-phase Navier-Stokes equations with surface tension in three dimensions. We use the open-source code Basilisk with adaptive mesh refinement in order to achieve very high effective resolutions of up to 4096^{3}, resolving length scales ranging from the breaker wavelength down to sizes comparable to the bubble Hinze scale. The resulting solutions are feature-rich, generating statistical ensembles of bubble and droplet populations along with velocity fields for analysis of turbulence. In addition to the bubble and droplet statistics, we also discuss the role of transverse instabilities in the breakup of the large volumes of entrained air and the concomitant transition of turbulence from two to three dimensions in the breaker development. Finally, we discuss matters of numerical grid convergence with respect to the bubble and droplet size distributions at the smallest scales.
]]>
Wouter Mostert et al.Inertial Energy Dissipation in Shallow-Water Breaking Waves
https://scholarsmine.mst.edu/mec_aereng_facwork/4506
https://scholarsmine.mst.edu/mec_aereng_facwork/4506Thu, 20 Aug 2020 11:01:23 PDT
We present direct numerical simulations of breaking solitary waves in shallow water to quantify the energy dissipation during the active breaking time. We find that this dissipation can be predicted by an inertial model based on Taylor's hypothesis as a function of the local wave height, depth and the beach slope. We obtain a relationship that gives the dissipation rate of a breaking wave on a shallow slope as a function of local breaking parameters. Next, we use empirical relations to relate the local wave parameters to the offshore conditions. This enables the energy dissipation to be predicted in terms of the initial conditions. We obtain good collapse of the numerical data with respect to the theoretical scaling.
]]>
Wouter Mostert et al.