Citation
Ward, Geoffrey M. (2011) The Simulation of Shock and ImpactDriven Flows with MieGrüneisen Equations of State. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8Q2QGT29. https://resolver.caltech.edu/CaltechTHESIS:12162010115725941
Abstract
An investigation of shock and impactdriven flows with MieGrüneisen equation of state derived from a linear shockparticle speed Hugoniot relationship is presented. Cartesian mesh methods using structured adaptive refinement are applied to simulate several flows of interest in an Eulerian frame of reference. The flows central to the investigation include planar RichtmyerMeshkov instability, the impact of a sphere with a plate, and an impactdriven Mach stem.
First, for multicomponent shockdriven flows, a dimensionally unsplit, spatially highorder, hybrid, centerdifference, limiter methodology is developed. Effective switching between centerdifference and upwinding schemes is achieved by a set of robust tolerance and Laxentropybased criteria [49]. Oscillations that result from such a mixed stencil scheme are minimized by requiring that the upwinding method approaches the centerdifference method in smooth regions. To attain this property a blending limiter is introduced based on the norm of the deviation of WENO reconstruction weights from ideal. The scheme is first demonstrated successfully for the linear advection equation in spatially fourth and sixthorder forms. A spatially fourthorder version of the method that combines a skewsymmetric kineticenergy preserving centerdifference scheme with a RoeRiemann solver is then developed and implemented in Caltech's Adaptive Mesh Refinement, Objectoriented C++ (AMROC) [16,17] framework for Euler flows.
The solver is then applied to investigate planar RichtmyerMeshkov instability in the context of an equation of state comparison. Comparisons of simulations with materials modeled by isotropic stress MieGrüneisen equations of state derived from a linear shockparticle speed Hugoniot relationship [36,52] to those of perfect gases are made with the intention of exposing the role of the equation of state. First, results for single and triplemode planar RichtmyerMeshkov instability between midocean ridge basalt (MORB) and molybdenum modeled by MieGrüneisen equations of state are presented for the case of a reflected shock. The singlemode case is explored for incident shock Mach numbers of 1.5 and 2.5. For the planar triplemode case a single incident Mach number of 2.5 is examined with the initial corrugation wave numbers related by k₁=k₂+k₃. A comparison is drawn to RichtmyerMeshkov instability in fluids with perfect gas equations of state utilizing matching of a nondimensional pressure jump across the incident shock, the postshock Atwood ratio, postshock amplitudetowavelength ratio, and time nondimensionalized by the Rcithmyer lineargrowth rate time constant prediction. Result comparison demonstrates difference in startup time and growth rate oscillations. Growth rate oscillation frequency is seen to correlate directly to the expected oscillation frequency of the transmitted and reflected shocks. For the singlemode cases, further comparison is given for vorticity distribution and corrugation centerline shortly after shock interaction that demonstrates only minor differences.
Additionally, examined is singlemode RichtmyerMeshkov instability when a reflected expansion wave is present for incident Mach numbers of 1.5 and 2.5. Comparison to perfect gas solutions in such cases yields a higher degree of similarity in startup time and growth rate oscillations. Vorticity distribution and corrugation centerline shortly after shock interaction is also examined. The formation of incipient weak shock waves in the heavy fluid driven by waves emanating from the perturbed transmitted shock is observed when an expansion wave is reflected.
Next, the ghost fluid method [83] is explored for application to impactdriven flows with MieGrüneisen equations of state in a vacuum. Free surfaces are defined utilizing a levelset approach. The levelset is reinitialized to the signed distance function periodically by solution to a HamiltonJacobi differential equation in artificial time. Flux reconstruction along each Cartesian direction of the domain is performed by subdividing in a way that allows for robust treatment of gridscale sized voids. Ghost cells in voided regions near the materialvacuum interface are determined from surfacenormal Riemann problem solution. The method is then applied to several impact problems of interest. First, a onedimensional impact problem is examined in MieGrüneisen aluminum with simple point erosion used to model separation by spallation under high tension. A similar threedimensional axisymmetric simulation of two rods impacting is then performed without a model for spallation. Further results for threedimensional axisymmetric simulation of a sphere hitting a plate are then presented.
Finally, a brief investigation of the assumptions utilized in modeling solids as isotropic fluids is undertaken. An Eulerian solver approach to handling elastic and elasticplastic solids is utilized for comparison to the simple fluid model assumption. First, in one dimension an impact problem is examined for elastic, elasticplastic, and fluid equations of state for aluminum. The results demonstrate that in one dimension the fluid models the plastic shock structure of the flow well. Further investigation is made using a threedimensional axisymmetric simulation of an impact problem involving a copper cylinder surrounded by aluminum. An aluminum slab impact drives a faster shock in the outer aluminum region yielding a Mach reflection in the copper. The results demonstrate similar plastic shock structures. Several differences are also notable that include a lack of rollup instability at the material interface and slipline emanating from the Mach stem's triple point.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  MieGrüneisen, Compressible flows, RichtmyerMeshkov instability, Leveset methods, Eulerian solid mechanics, Hyperbolic conservation laws, Mach stem 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Aeronautics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Group:  GALCIT 
Thesis Committee: 

Defense Date:  3 December 2010 
Record Number:  CaltechTHESIS:12162010115725941 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:12162010115725941 
DOI:  10.7907/8Q2QGT29 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  6211 
Collection:  CaltechTHESIS 
Deposited By:  Geoff Ward 
Deposited On:  23 Dec 2010 19:30 
Last Modified:  09 Oct 2019 17:07 
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