A dynamically adaptive multigrid algorithm for the incompressible Navier-Stokes equations: Validation and model problems

An algorithm is described for the solution of the laminar, incompressible Navier-Stokes equations. The basic algorithm is a multigrid based on a robust, box-based smoothing step. Its most important feature is the incorporation of automatic, dynamic mesh refinement. This algorithm supports generalized simple domains. The program is based on a standard staggered-grid formulation of the Navier-Stokes equations for robustness and efficiency. Special grid transfer operators were introduced at grid interfaces in the multigrid algorithm to ensure discrete mass conservation. Results are presented for three models: the driven-cavity, a backward-facing step, and a sudden expansion/contraction

Direct Coupling Method for Time-Accurate Solution of Incompressible Navier-Stokes Equations

A noniterative finite difference numerical method is presented for the solution of the incompressible Navier-Stokes equations with second order accuracy in time and space. Explicit treatment of convection and diffusion terms and implicit treatment of the pressure gradient give a single pressure Poisson equation when the discretized momentum and continuity equations are combined. A pressure boundary condition is not needed on solid boundaries in the staggered mesh system. The solution of the pressure Poisson equation is obtained directly by Gaussian elimination. This method is tested on flow problems in a driven cavity and a curved duct

Application of a fractional-step method to incompressible Navier-Stokes equation

A numerical method for computing three dimensional, time dependent incompressible flows is presented. The method is based on a fractional step, or time-splitting, scheme in conjunction with the approximate-factorization technique. The use of velocity boundary conditions for the intermediate velocity field leads to inconsistent numerical solutions. Appropriate boundary conditions for the intermediate velocity field are derived and tested. Numerical solutions for flow inside a driven cavity and over a backward-facing step are presented and compared with experimenal data and other numerical results

Analysis and optimization of film cooling effectiveness

In the first part, an optimization strategy is described that combines high-fidelity simu- lations with response surface construction, and is applied to pulsed film cooling for turbine blades. The response surface is constructed for the film cooling effectiveness as a function of duty cycle, in the range of DC between 0.05 and 1, and pulsation frequency St in the range of 0.2-2, using a pseudo-spectral projection method. The jet is fully modulated and the blowing ratio, when the jet is on, is 1.5 in all cases. Overall 73 direct numerical sim- ulations (DNS) using spectral element method were performed to sample the film cooling effectiveness on a Clenshaw-Curtis grid in the design space. It is observed that in the parameter space explored a global optimum exists, and in the present study, the best film cooling effectiveness is found at DC = 0.14 and St = 1.03. In the same range of DC and St, four other local optimums were found. The gradient-based optimization algorithms are argued to be unsuitable for the current problem due to the non-convexity of the objective function. In the second part, the effect of randomness of blowing ratio on film cooling performance is investigated by combining direct numerical simulations with a stochastic collocation ap- proach. The blowing ratio variations are assumed to have a truncated Gaussian distribution with mean of 0.3 and the standard variation of approximately 0.1. The parametric space is discretized using Multi-Element general Polynomial Chaos (ME-gPC) with five elements where general polynomial chaos of order 3 is used in each element. Direct numerical simula- tions were carried out using spectral/hp element method to sample the governing equations in space and time. The probability density function of the film cooling effectiveness was obtained and the standard deviation of the adiabatic film cooling effectiveness on the blade surface was calculated. A maximum standard deviation of 15% was observed in the region within a four-jet-diameter distance downstream of the exit hole. The spatially-averaged adiabatic film cooling effectiveness was 0.23 0.02. The calculation of all the statistical properties were carried out as off-line post-processing. Overall the computational strategy is shown to be very effective with the total computational cost being equivalent to solving twenty independent direct numerical simulations that are performed concurrently. In the third part, an accurate and efficient finite difference method for solving the incompressible Navier-Stokes equations on curvilinear grids is developed. This method combines the favorable features of the staggered grid and semi-staggered grid approaches. A novel symmetric finite difference discretization of the Poisson-Neumann problem on curvilinear grids is also presented. The validity of the method is demonstrated on four benchmark problems. The Taylor-Green vortex problem is solved on a curvilinear grid with highly skewed cells and a second-order convergence in .-norm is observed. The mixed convection in a lid-driven cavity is solved on a highly curvilinear grid and excellent agreement with literature is obtained. The results for flow past a cylinder are compared with the existing experimental data in the literature. As the fourth case, three dimensional time-dependent incompressible flow in a curved tube is solved. The predictions agree well with the measured data, and validate the approach used

Arbitrary Lagrangian-Eulerian form of flowfield dependent variation (ALE-FDV) method for moving boundary problems

Flowfield Dependent Variation (FDV) method is a mixed explicit-implicit numerical scheme that was originally developed to solve complex flow problems through the use of so-called implicitness parameters. These parameters determine the implicitness of FDV method by evaluating local gradients of physical flow parameters, hence vary across the computational domain. The method has been used successfully in solving wide range of flow problems. However it has only been applied to problems where the objects or obstacles are static relative to the flow. Since FDV method has been proved to be able to solve many complex flow problems, there is a need to extend FDV method into the application of moving boundary problems where an object experiences motion and deformation in the flow. With the main objective to develop a robust numerical scheme that is applicable for wide range of flow problems involving moving boundaries, in this study, FDV method was combined with a body interpolation technique called Arbitrary Lagrangian-Eulerian (ALE) method. The ALE method is a technique that combines Lagrangian and Eulerian descriptions of a continuum in one numerical scheme, which then enables a computational mesh to follow the moving structures in an arbitrary movement while the fluid is still seen in a Eulerian manner. The new scheme, which is named as ALE-FDV method, is formulated using finite volume method in order to give flexibility in dealing with complicated geometries and freedom of choice of either structured or unstructured mesh. The method is found to be conditionally stable because its stability is dependent on the FDV parameters. The formulation yields a sparse matrix that can be solved by using any iterative algorithm. Several benchmark stationary and moving body problems in one, two and three-dimensional inviscid and viscous flows have been selected to validate the method. Good agreement with available experimental and numerical results from the published literature has been obtained. This shows that the ALE-FDV has great potential for solving a wide range of complex flow problems involving moving bodies

A review of urban roughness sublayer turbulence

It is becoming increasingly important that we can understand and model flow processes in urban areas. Applications such as weather forecasting, air quality and sustainable urban development rely on accurate modelling of the interface between an urban surface and the atmosphere above. This review gives an overview of current understanding of turbulence generated by an urban surface up to a few building heights, the layer called the roughness sublayer (RSL). High quality datasets are also identified which can be used in the development of suitable parameterisations of the urban RSL. Datasets derived from physical and numerical modelling, and full-scale observations in urban areas now exist across a range of urban-type morphologies (e.g. street canyons, cubes, idealised and realistic building layouts). Results show that the urban RSL depth falls within 2 – 5 times mean building height and is not easily related to morphology. Systematic perturbations away from uniform layouts (e.g. varying building heights) have a significant impact on RSL structure and depth. Considerable fetch is required to develop an overlying inertial sublayer, where turbulence is more homogeneous, and some authors have suggested that the “patchiness” of urban areas may prevent inertial sublayers from developing at all. Turbulence statistics suggest similarities between vegetation and urban canopies but key differences are emerging. There is no consensus as to suitable scaling variables, e.g. friction velocity above canopy vs. square root of maximum Reynolds stress, mean vs. maximum building height. The review includes a summary of existing modelling practices and highlights research priorities

Towards the study of flying snake aerodynamics, and an analysis of the direct forcing method

Immersed boundary methods are a class of techniques in computational fluid dynamics where the Navier-Stokes equations are simulated on a computational grid that does not conform to the interfaces in the domain of interest. This facilitates the simulation of flows with complex moving and deforming geometries without considerable effort wasted in generating the mesh. The first part of this dissertation is concerned with the aerodynamics of the cross-section of a species of flying snake, Chrysopelea paradisi (paradise tree snake). Past experiments have shown that the unique cross-section of this snake, which can be described as a lifting bluff body, produces an unusual lift curve--with a pronounced peak in lift coefficient at an angle of attack of 35 degrees for Reynolds numbers 9000 and beyond. We studied the aerodynamics of the cross-section using a 2-D immersed boundary method code. We were able to qualitatively reproduce the spike in the lift coefficient at the same angle of attack for flows beyond a Reynolds number of 2000. This phenomenon was associated with flow separation at the leading edge of the body that did not result in a stall. This produced a stronger vortex and an associated reduction in pressure on the dorsal surface of the snake cross-section, which resulted in higher lift. The second part of this work deals with the analysis of the direct forcing method, which is a popular immersed boundary method for flows with rigid boundaries. We begin with the fully discretized Navier-Stokes equations along with the appropriate boundary conditions applied at the solid boundary, and derive the fractional step method as an approximate block LU decomposition of this system. This results in an alternate formulation of the direct forcing method that takes into consideration mass conservation at the immersed boundaries and also handles the pressure boundary conditions more consistently. We demonstrate that this method is between first and second-order accurate in space when linear interpolation is used to enforce the boundary conditions on velocity. We then develop a theory for the order of accuracy of the direct forcing method with linear interpolation. For a simple 1-D case, we show that the method can converge at a range of rates for different locations of the solid body with respect to the mesh. But this effect averages out in higher dimensions and results in a scheme that has the same order of accuracy as the expected order of accuracy of the interpolation at the boundary. The discrete direct forcing method for the Navier-Stokes equations exhibits an order of accuracy between 1 and 2 because the velocities at the boundary are linearly interpolated, but the resulting boundary conditions on the pressure gradient turn out to be only first-order accurate. We recommend linearly interpolating the pressure gradient as well to make the method fully second-order accurate. We have also developed two open source codes in the course of these studies. The first, cuIBM, is a two-dimensional immersed boundary method code that runs on a single GPU. It can simulate incompressible flow around rigid bodies with prescribed motion. It is based on the general idea of a fractional step method as an approximate block LU decomposition, and can incorporate any type of immersed boundary method that can be made to fit within this framework. The second code, PetIBM, can simulate both two and three-dimensional incompressible flow and runs in parallel on multiple CPUs. Both codes have been validated using well-known test cases

Microstructural-Based Modeling Framework for High Temperature Behavior of Ferritic-Martensitic Steels Using Crystal Plasticity and Grain Boundary Finite Element Approaches

Ferritic/martensitic 9-12Cr steel alloys, have had widespread use as structural materials in power plants. Among this family of alloys, Grade 91 (Gr91) steel was a landmark in the development of 9-12Cr alloys. However, the unique microstructure complexity of the alloy has raised doubt regarding the techniques of data extrapolation in estimating its service-life for operation in next-generation power plants at higher temperatures and presssures. Conservatism becomes essential when the alloy is to be used in components lasting the life-cycle of power plants without replacement.This dissertation develops a physically-based microstructural model for creep rupture at 600 degrees Celsius for Gr91 steel as well as fundamental modeling tools that apply more broadly to microstructural modeling in metals. Key features of the Gr91 modeling framework capture the mechanical behavior of its prior austenite grains (PAG) and grain boundaries. Ultimately, a constitutive expression was adopted that captured the response from experiments conducted in the creep strain rate regime.An initial model intended to simulate low-cycle fatigue was first developed using the idea of geometrically necessary dislocations (GNDs) in crystal plasticity (CP) framework. That necessitated evaluating strain gradients and a patch-recovery method was implemented to recover a linear elastic deformation gradient field across the domain in linear elements. A Lie-group to Lie-algebra mapping was used to preserve orthogonality when projecting the rotation tensor from the elements’ Gauss points to the nodes.A statistically-stored dislocation density model was investigated to span the regimes of moderate strain rates (tension tests) to low strain rates (creep tests). Calibration of this model was possible against tension tests, but its application to creept tests suggested that other dislocation mechanisms were present during the primary creep regime of Gr91. Therefore, the CP model in the PAGs was changed to represent dislocation climb-glide motion and recovery along with linear viscous diffusional creep for point defect diffusion. This revised model more closely captured the measurements of creep response.Lastly, a robust Discontinuous Galerkin method is proposed to model the grain boundary interface elements to address traction oscillations observed for cohesive models. Stability and convergence are assessed along with non-conforming meshes

An Asymptotic-Preserving all-speed scheme for the Euler and Navier-Stokes equations

We present an Asymptotic-Preserving 'all-speed' scheme for the simulation of compressible flows valid at all Mach-numbers ranging from very small to order unity. The scheme is based on a semi-implicit discretization which treats the acoustic part implicitly and the convective and diffusive parts explicitly. This discretization, which is the key to the Asymptotic-Preserving property, provides a consistent approximation of both the hyperbolic compressible regime and the elliptic incompressible regime. The divergence-free condition on the velocity in the incompressible regime is respected, and an the pressure is computed via an elliptic equation resulting from a suitable combination of the momentum and energy equations. The implicit treatment of the acoustic part allows the time-step to be independent of the Mach number. The scheme is conservative and applies to steady or unsteady flows and to general equations of state. One and Two-dimensional numerical results provide a validation of the Asymptotic-Preserving 'all-speed' properties

Thermophysical Phenomena in Metal Additive Manufacturing by Selective Laser Melting: Fundamentals, Modeling, Simulation and Experimentation

Among the many additive manufacturing (AM) processes for metallic materials, selective laser melting (SLM) is arguably the most versatile in terms of its potential to realize complex geometries along with tailored microstructure. However, the complexity of the SLM process, and the need for predictive relation of powder and process parameters to the part properties, demands further development of computational and experimental methods. This review addresses the fundamental physical phenomena of SLM, with a special emphasis on the associated thermal behavior. Simulation and experimental methods are discussed according to three primary categories. First, macroscopic approaches aim to answer questions at the component level and consider for example the determination of residual stresses or dimensional distortion effects prevalent in SLM. Second, mesoscopic approaches focus on the detection of defects such as excessive surface roughness, residual porosity or inclusions that occur at the mesoscopic length scale of individual powder particles. Third, microscopic approaches investigate the metallurgical microstructure evolution resulting from the high temperature gradients and extreme heating and cooling rates induced by the SLM process. Consideration of physical phenomena on all of these three length scales is mandatory to establish the understanding needed to realize high part quality in many applications, and to fully exploit the potential of SLM and related metal AM processes