# Browsing by Subject "Convection-diffusion"

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Item A DPG method for convection-diffusion problems(2013-08) Chan, Jesse L.; Demkowicz, Leszek; Moser, Robert deLanceyShow more Over the last three decades, CFD simulations have become commonplace as a tool in the engineering and design of high-speed aircraft. Experiments are often complemented by computational simulations, and CFD technologies have proved very useful in both the reduction of aircraft development cycles, and in the simulation of conditions difficult to reproduce experimentally. Great advances have been made in the field since its introduction, especially in areas of meshing, computer architecture, and solution strategies. Despite this, there still exist many computational limitations in existing CFD methods; in particular, reliable higher order and hp-adaptive methods for the Navier-Stokes equations that govern viscous compressible flow. Solutions to the equations of viscous flow can display shocks and boundary layers, which are characterized by localized regions of rapid change and high gradients. The use of adaptive meshes is crucial in such settings -- good resolution for such problems under uniform meshes is computationally prohibitive and impractical for most physical regimes of interest. However, the construction of "good" meshes is a difficult task, usually requiring a-priori knowledge of the form of the solution. An alternative to such is the construction of automatically adaptive schemes; such methods begin with a coarse mesh and refine based on the minimization of error. However, this task is difficult, as the convergence of numerical methods for problems in CFD is notoriously sensitive to mesh quality. Additionally, the use of adaptivity becomes more difficult in the context of higher order and hp methods. Many of the above issues are tied to the notion of robustness, which we define loosely for CFD applications as the degradation of the quality of numerical solutions on a coarse mesh with respect to the Reynolds number, or nondimensional viscosity. For typical physical conditions of interest for the compressible Navier-Stokes equations, the Reynolds number dictates the scale of shock and boundary layer phenomena, and can be extremely high -- on the order of 10⁷ in a unit domain. For an under-resolved mesh, the Galerkin finite element method develops large oscillations which prevent convergence and pollute the solution. The issue of robustness for finite element methods was addressed early on by Brooks and Hughes in the SUPG method, which introduced the idea of residual-based stabilization to combat such oscillations. Residual-based stabilizations can alternatively be viewed as modifying the standard finite element test space, and consequently the norm in which the finite element method converges. Demkowicz and Gopalakrishnan generalized this idea in 2009 by introducing the Discontinous Petrov-Galerkin (DPG) method with optimal test functions, where test functions are determined such that they minimize the discrete linear residual in a dual space. Under the ultra-weak variational formulation, these test functions can be computed locally to yield a symmetric, positive-definite system. The main theoretical thrust of this research is to develop a DPG method that is provably robust for singular perturbation problems in CFD, but does not suffer from discretization error in the approximation of test functions. Such a method is developed for the prototypical singular perturbation problem of convection-diffusion, where it is demonstrated that the method does not suffer from error in the approximation of test functions, and that the L² error is robustly bounded by the energy error in which DPG is optimal -- in other words, as the energy error decreases, the L² error of the solution is guaranteed to decrease as well. The method is then extended to the linearized Navier-Stokes equations, and applied to the solution of the nonlinear compressible Navier-Stokes equations. The numerical work in this dissertation has focused on the development of a 2D compressible flow code under the Camellia library, developed and maintained by Nathan Roberts at ICES. In particular, we have developed a framework allowing for rapid implementation of problems and the easy application of higher order and hp-adaptive schemes based on a natural error representation function that stems from the DPG residual. Finally, the DPG method is applied to several convection diffusion problems which mimic difficult problems in compressible flow simulations, including problems exhibiting both boundary layers and singularities in stresses. A viscous Burgers' equation is solved as an extension of DPG to nonlinear problems, and the effectiveness of DPG as a numerical method for compressible flow is assessed with the application of DPG to two benchmark problems in supersonic flow. In particular, DPG is used to solve the Carter flat plate problem and the Holden compression corner problem over a range of Mach numbers and laminar Reynolds numbers using automatically adaptive schemes, beginning with very under-resolved/coarse initial meshes.Show more Item Scaling of solutal convection in porous media(2017-12-08) Liang, Yu, active 21st century; DiCarlo, David Anthony, 1969-; Hesse, Marc; Lake, Larry; Mohanty, Kishore; Balhoff, MatthewShow more Convective dissolution trapping is an important mechanism for CO₂ mitigation because of its high security and long-term storage capacity. This process is a problem of solutal convection in porous media, which is a classic example of symmetry breaking and pattern formation. The convective solute flux and geometry of the convective pattern are thought to be controlled by the molecular Rayleigh number, Raₘ, i.e., the ratio of the buoyant driving forces over diffusive dissipation. The dimensionless convective solute flux, Sh (Sherwood number), is thought to increase with Raₘ approximately linearly, as Sh ~ Raₘ. The spacing of the convective fingers, δ, relative to the domain height, H, is thought to decrease approximately as [mathematical equation]. However, there is little experimental verification of these fundamental scaling laws for solutal convection in porous media. To understand the controlling physics of CO₂ convective dissolution in aquifers and verify relevant fundamental scaling laws, I conduct convective dissolution experiments using analog fluids in a porous medium. By changing the controlling parameters, including permeability and maximum density difference, corresponding convective velocity, dissolution flux, and finger pattern are measured for each combination. The experimental results shows that these fundamental scaling laws do not hold for the experiments in porous media composed of glass beads. Instead, I observe that the dissolution flux levels off as Raₘ increases and that the finger spacing increases rather than decreases with increasing Raₘ. The classic scaling analysis breaks down because it does not consider the dominant dissipative mechanism in porous media, mechanical dispersion. Its influence on convection is captured by a dispersive Rayleigh number, Ra [subscript d] = H / αT, where αT is the transverse dispersivity. In experimental studies, dispersion dominates molecular diffusion, i.e., Ra [subscript d] ≤ Raₘ and therefore selects the finger spacing. Increasing the bead size of the porous medium increases Raₘ but decreases Ra [subscript d], leading to a coarsening of the convective pattern. The dissolution flux is controlled by Raₘ, which captured the buoyant driving force in the convection. However, the inherent anisotropy of mechanical dispersion leads to an asymmetry in the convective pattern that eventually limits the dissolution flux in the high Raₘ limit, resulting in the breakdown of the classic convective solute flux scaling. This anisotropy induced asymmetry and corresponding flux reduction are verified by a numerical simulation study. The results show that mechanical dispersion, which was ignored before, plays an important role in quantifying solutal convection in porous media. Since dispersivity generally increases with the scale of observations, knowledge obtained from the counter-intuitive results can be applied to predict mass transfer in large-scale applications such as CO₂ convective dissolution storage in aquifers.Show more