W. Roger Briley

Numerical method for predicting three-dimensional steady viscous flow in ducts

Abstract A numerical method for predicting three-dimensional, steady viscous flow in ducts is described. The method utilizes approximate governing equations which are applicable to flows having strong convection in one primary flow direction. The governing equations require a coordinate system as input to define primary and secondary flow directions, and an inviscid first approximation to static pressure gradients arising from curved flow geometries. The equations are parabolic and are solved by stepwise integration in the primary flow direction from prescribed upstream initial conditions. Specific details of the method are given by way of application to a special case chosen for its simplicity, that of laminar flow in the entrance region of straight rectangular ducts. A numerical method based on an alternating-direction implicit (ADI) scheme is described and used to compute solutions for flow in ducts having aspect ratios of 1:1 and 2:1; in one case, the effect of thermal convection caused by a transverse buoyancy force is also included. The computed solutions are found to be in good agreement with experimental velocity-profile and pressure-drop measurements. Extensions to treat more general geometries and to include compressibility effects and turbulent transport processes are possible and seem warranted by the present results.

A numerical study of laminar separation bubbles using the Navier-Stokes equations

The flow in a two-dimensional laminar separation bubble is analyzed by means of finite-difference solutions to the Navier-Stokes equations for incompressible flow. The study was motivated by the need to analyze high-Reynolds-number flow fields having viscous regions in which the boundary-layer assumptions are questionable. The approach adopted in the present study is to analyze the flow in the immediate vicinity of the separation bubble using the Navier-Stokes equations. It is assumed that the resulting solutions can then be patched to the remainder of the flow field, which is analyzed using boundary-layer theory and inviscid-flow analysis. Some of the difficulties associated with patching the numerical solutions to the remainder of the flow field are discussed, and a suggestion for treating boundary conditions is made which would permit a separation bubble to be computed from the Navier-Stokes equations using boundary conditions from inviscid and boundary-layer solutions without accounting for interaction between individual flow regions. Numerical solutions are presented for separation bubbles having Reynolds numbers (based on momentum thickness) of the order of 50. In these numerical solutions, separation was found to occur without any evidence of the singular behaviour at separation found in solutions to the boundary-layer equations. The numerical solutions indicate that predictions of separation by boundary-layer theory are not reliable for this range of Reynolds number. The accuracy and validity of the numerical solutions are briefly examined. Included in this examination are comparisons between the Howarth solution of the boundary-layer equations for a linearly retarded freestream velocity and the corresponding numerical solutions of the Navier-Stokes equations for various Reynolds numbers.

An overview and generalization of implicit Navier–Stokes algorithms and approximate factorization

Abstract A theme of linearization and approximate factorization provides the context for a retrospective overview of the development and evolution of implicit numerical methods for the compressible and incompressible Euler and Navier–Stokes algorithms. This topic was chosen for this special volume commemorating the recent retirements of R.M. Beam and R.F. Warming. A generalized treatment of approximate factorization schemes is given, based on an operator notation for the spatial approximation. The generalization focuses on the implicit structure of Euler and Navier–Stokes algorithms as nonlinear systems of partial differential equations, with details of the spatial approximation left to operator definitions. This provides a unified context for discussing noniterative and iterative time-linearized schemes, and Newton iteration for unsteady nonlinear schemes. The factorizations include alternating direction implicit, LU and line relaxation schemes with either upwind or centered spatial approximations for both compressible and incompressible flows. The noniterative schemes are best suited for steady flows, while the iterative schemes are well suited for either steady or unsteady flows. This generalization serves to unify a large number of schemes developed over the past 30 years.

Aerosol Propagation in an Urban Environment

The objective of this study is to demonstrate the capability of an arbitrary mach number algorithm to predict aerosol propagation in an urban environment. A preconditioned approach is applied to an unstructured mesh to determine accurately the highly unsteady turbulent flow field about the urban setting. DES modifications are implemented into a hybrid k − , k − ω turbulence model and evaluated. A scalar transport model is used to release and to advect the aerosol agent through the urban landscape. Comparisons between RANS and DES turbulence modeling are presented for multiple agent release scenarios.

Parallel solution of viscous incompressible flow on multi-block structured grids using MPI

Publisher Summary The aim of this chapter is to develop a parallel code based on an existing unsteady three-dimensional incompressible viscous flow solver for simulation of flows past complex configurations using multi-block structured grids and using message passing interface (MPI) for message passing. The linearized implicit solution algorithm used is modified for execution in parallel using a block-decoupled sub-iterative strategy, and a heuristic performance estimate is developed to guide the parallel problem definition. The chapter provides extensive evaluations of both algorithmic and parallel performance for a test problem consisting of axisymmetric flow past an unappended submarine hull with a Reynolds number of 12 million. Other results are given for three-dimensional appended submarine cases having 0.6M and 3.3M grid points. The decoupled sub-iteration algorithm allows the convergence rate of the sequential algorithm to be recovered at reasonable cost by using a sufficient number of sub-iterations, and allows decompositions having multiple subdivisions across boundary layers. The chapter concludes that the present approach gives reasonable parallel performance for large scale steady flow simulations wherein the grid and decomposition are appropriately sized for the targeted computer architecture.

A Parabolic Method without Pressure Approximations for Wind Turbines

A Parabolized Navier-Stokes (PNS) approximation for the incompressible equations is developed to simulate flows through a wind turbine. The PNS formulation differs from commonly used parabolic formulations in that the pressure field is calculated as a dependent variable without any approximation for the streamwise pressure gradient. The turbine influence is modeled by incorporating time-averaged aerodynamic forces predicted by an Actuator Line model (FAST developed at NREL). These forces are time averaged and incorporated as localized source terms near the turbine. Using this coupled time-averaged spatial-marching method, the calculation can begin well upstream of the turbine and continue through the turbine to predict the entire wake region. Laminar and turbulent flatplate boundary-layer cases are used here for basic validation of the PNS Algorithm. Computed solutions for the NREL offshore 5-MW baseline wind turbine are compared with the blade-resolved Navier-Stokes solutions for verification.

A Parabolic Method for Accurate and Efficient Wind Farm Simulation

A Parabolized Navier-Stokes (PNS) approximation for the incompressible equations is developed and utilized to simulate flow through a wind farm of five wind turbines. The PNS formulation differs from commonly used parabolic formulations in that the pressure field is calculated as a dependent variable without any approximation for the streamwise pressure gradient. The wind turbine is modeled by incorporating time-averaged aerodynamic forces predicted by an Actuator Line model (FAST developed at NREL). Using this coupled timeaveraged spatial-marching method, the calculation can begin well upstream of the wind farm and continue through the turbines to predict the entire flow field including the wakes. The model was verified for wind turbines by comparing the computed solutions for the NREL offshore 5-MW baseline wind turbine with the blade-resolved Navier-Stokes solutions. Very good agreement was obtained and the runtime on a single desktop computer was less than 80 minutes. Results for the wind farm are presented. The simulation on a desktop computer took 271 minutes.

Extension of a Parabolic Method without Pressure Approximations for Wind Turbines in ABL Flows

A Parabolized Navier-Stokes (PNS) approximation for the incompressible equations is developed to simulate flows through a wind turbine embedded in an atmospheric boundary layer (ABL). The PNS formulation differs from commonly used parabolic formulations in that the pressure field is calculated as a dependent variable without any approximation for the streamwise pressure gradient. The turbine influence is modeled by incorporating timeaveraged aerodynamic forces predicted by an actuator-line model (FAST developed at NREL). Using this coupled time-averaged spatial-marching method, the calculation can begin well upstream of the turbine and continue through the turbine to predict the entire wake region. In this study, the original model is extended to simulate neutral atmospheric boundary layers (ABL) and study the wake development of a single wind turbine. The PNS model has been validated for laminar and turbulent flat-plate boundary-layer cases and verified for a wind turbine (uniform inflow conditions) by comparing the computed solutions with the blade-resolved Navier-Stokes solutions for the NREL offshore 5-MW baseline wind turbine.

Calculation of three-dimensional turbulent subsonic flows in transition ducts

A method for computing three-dimensional turbulent subsonic flow in curved ducts is being developed. A set of tube-like surface oriented coordinates is employed for a general class of geometries applicable to subsonic diffusers with offset bends. The geometric formulation is complex and no previous treatment of this class of viscous flow problems is known to the authors. The duct centerline is a space curve specified by piecewise polynomials. A Frenet frame is located on the centerline at each axial location. The cross sections are described by super-ellipses imbedded in the Frenet frame. Duct surfaces are also coordinate surfaces, which greatly simplifies the boundary conditions. The resulting coordinates are nonorthogonal.