An adaptive moving mesh method for forced curve shortening flow

We propose a novel adaptive moving mesh method for the numerical solution of a forced curve shortening geometric evolution equation. Control of the mesh quality is obtained using a tangential mesh velocity derived from a mesh equidistribution principle, where a positive adaptivity measure or monitor function is approximately equidistributed along the evolving curve. Central finite differences are used to discretise in space the governing evolution equation for the position vector and a second-order implicit scheme is used for the temporal integration. Simulations are presented indicating the generation of meshes which resolve areas of high curvature and are of second-order accuracy. Furthermore, the new method delivers improved solution accuracy compared to the use of uniform arc-length meshes

Curve shortening flow coupled to lateral diffusion

We present and analyze a semi-discrete finite element scheme for a system consisting of a geometric evolution equation for a curve and a parabolic equation on the evolving curve. More precisely, curve shortening flow with a forcing term that depends on a field defined on the curve is coupled with a diffusion equation for that field. The scheme is based on ideas of [Dziuk, G. Discrete anisotropic curve shortening flow, SIAM J. Numer. Anal. 36, 6 (1999), 1808–1830] for the curve shortening flow and [Dziuk, G., and Elliott, C. M. Finite elements on evolving surfaces, IMA J. Numer. Anal. 27, 2 (2007), 262–292] for the parabolic equation on the moving curve. Additional estimates are required in order to show convergence, most notably with respect to the length element: While in [Dziuk, G. Discrete anisotropic curve shortening flow, SIAM J. Numer. Anal. 36, 6 (1999), 1808–1830] an estimate of its error was sufficient we here also need to estimate the time derivative of the error which arises from the diffusion equation. Numerical simulation results support the theoretical findings

A computational method for the coupled solution of reaction–diffusion equations on evolving domains and manifolds: application to a model of cell migration and chemotaxis

In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk–surface reaction–diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk–surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane

Examination of Flow Dynamics and Passive Cooling in an Ultra Compact Combustor

The Ultra Compact Combustor (UCC) promises to greatly reduce the size of a gas turbine engine’s combustor by altering the manner in which fuel is burnt. Differing from the common axial flow combustor, the UCC utilizes a rotating flow, coaxial to the engine’s primary axis, in an outboard circumferential cavity as the primary combustion zone. The present study investigates two key UCC facets required to further this combustor design. The first area of investigation is cooling of the Hybrid Guide Vane (HGV). This UCC specific hardware acts as a combustor center body that alters the exit flow angle and acts as a secondary combustion zone. As improved UCC designs yield higher operating temperatures, cooling of this component must be considered. Previous numeric efforts determined the viability of a passive cooling scheme where cooler compressor air was drawn into an opening at the stagnation region of the HGV and used for both internal and film-cooling of the vane. The present study experimentally investigated the performance of five cooled HGV configurations, each having a unique combination of film-cooling, internal passage area, and internal passage geometry, over a range of flow conditions. Results confirmed the efficacy of HGV cooling via passive air ingestion. The internal vane geometry affected both internal coolant mass flow and pressure rise offering a means to adjust both parameters based on cooling requirements. Vane internal to external pressure differential had a strong impact on blowing ratio where low differences provided sub-optimal cooling benefits and large differences caused the coolant jet to penetrate into the freestream, altering its flow structure. Next, the UCC hardware was modified to accommodate enhanced combustor diagnostics. Previous studies relied on point measurements or optical analysis of a small portion of the cavity greatly limiting the obtainable data. By replacing the rear enclosure of the combustor with a clear quartz back plate, analysis of the flow dynamics in the entire cavity was made possible. Design features that ensured a sealed combustor while still accommodating thermal expansion of the quartz allowed extensive data to be collected while avoiding damage to the modified hardware. These modifications were then leveraged to assess the underlying complex aerodynamic and combustion phenomenon driving previously observed average combustor exit temperatures. Emissions of OH and CH radicals were recorded using intensified relay optics and a high speed camera providing information on flame location. Tracking the flame’s movement enabled its velocity within the cavity to be determined. Observed average cavity tangential velocity increased with cavity airflow rate and had the highest local value within the cavity, near the step air injection. Velocity was lowest at the vane tips of the HGV, which locally disrupted flow migration. Increased tangential velocities adversely affected the ability of the flame stabilizing mechanism to anchor a flame, resulting in less overall flame activity at higher cavity velocities. This was primarily driven by the method of air injection which, at all but the lowest tangential velocities, prevented flow recirculation in half of the stabilizing zones. Further increase in cavity tangential velocity altogether pushed the flame off of the outside diameter of the combustor, which temporarily stabilized around the HGV prior to extinguishing. Lastly, analysis of combustion event movement indicated larger tangential velocities decreased residence time coinciding with a previously observed drop in average combustor exit temperature

A computational method for the coupled solution of reaction-diffusion equations on evolving domains and manifolds : application to a model of cell migration and chemotaxis

In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk-surface reaction-diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk-surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane

Enhanced Flow Migration in Full Annular Ultra Compact Combustor

Since combustion efficiency in modern jet engines has stabilized, attention has turned to improving the combustor by improving the thrust-to-weight ratio. The Ultra Compact Combustor (UCC) is a means to reduce the weight of the combustor while ensuring exhaust meets increasingly stringent government emission standards. Combustion occurs within the UCC under a g-load in the circumferential direction, which maintains combustion efficiency while decreasing axial combustor length. Previous analysis optimized the combustion chamber flame characteristics with a common upstream air source. Previously, issues for the UCC were inspired by integration into a traditional axial turbojet. The focus of this investigation was to increase migration of the hot combustion products to the middle of the hybrid vane’s exit plane. This was done by varying the dimensions of the UCC combustion cavity, the air driver configuration into the cavity, as well as adding a radial vane cavity into the center-body guide vanes. In order to accomplish this, a temperature measurement collection technique called thin filament pyrometry was implemented to obtain high fidelity data. Also, the AFIT UCC required an accurate initial emissions baseline to be established; this baseline consisted of collecting five different gaseous species for each considered geometry. These data points were then compared against each other and previously collected temperature values. Optimal exit efficiency and temperature profiles were obtained through modifications to the hybrid vane passage and the air driver geometry