Nonoscillatory Forward-in-time Research Articles

Explicit and implicit large eddy simulations of variable-density low-speed flows

The application of the Non-oscillatory Forward-in-Time (NFT) class of solvers to facilitate conventional explicit Large Eddy Simulations (LES) and implicit LES (ILES) of low-speed turbulent flow encompassing substantial local density variations is explored. Schemes solving two different sets of Low Mach Number (LMN) equations are proposed with the aim of capturing variable-density effects arising from either a non-isothermal distribution in single-species flow or compositional variation occurring in a binary-species mixing under isothermal conditions. Both schemes employ the semi-implicit NFT integrators based on the Multidimensional Positive Definite Advection Transport Algorithm and a non-symmetric Krylov subspace solver for the variable-coefficient Poisson problems arising from different treatments applied to the non-zero velocity divergence term in the two sets of the LMN equations. The developments are successfully validated using three diverse test cases, differentially heated cavity flow, non-isothermal free-jets, and a helium plume. The reported simulations show that both the dynamic Smagorinsky model in LES and ILES subgrid-scale treatments yield similar accuracy in capturing turbulent effects for the considered flows.

Stratified flow past a sphere at moderate Reynolds numbers

A numerical study of stably stratified flows past spheres at Reynolds numbers Re=200 and Re=300 is reported. In these flow regimes, a neutrally stratified laminar flow induces distinctly different near-wake features. However, the flow behaviour changes significantly as the stratification increases and suppresses the scale of vertical displacements of fluid parcels. Computations for a range of Froude numbers Fr∈[0.1,∞] show that as Froude number decreases, the flow patterns for both Reynolds numbers become similar. The representative simulations of the lee-wave instability at Fr=0.625 and the two-dimensional vortex shedding at Fr=0.25 regimes are illustrated for flows past single and tandem spheres, thereby providing further insight into the dynamics of stratified flows past bluff bodies. In particular, the reported study examines the relative influence of viscosity and stratification on the dividing streamline elevation, wake structure and flow separation. The solutions of the Navier-Stokes equations in the incompressible Boussinesq limit are obtained on unstructured meshes suitable for simulations involving multiple bodies. Computations are accomplished using the finite volume, non-oscillatory forward-in-time (NFT) Multidimensional Positive Definite Transport Algorithm (MPDATA) based solver. The impact and validity of the numerical approximations, especially for the cases exhibiting strong stratification, are also discussed. Qualitative and quantitative comparisons with available laboratory experiments and prior numerical studies confirm the validity of the numerical approach.

Non-oscillatory forward-in-time integrators for viscous incompressible flows past a sphere

A non-oscillatory forward-in-time (NFT) integrator is developed to provide solutions of the Navier–Stokes equations for incompressible flows. Simulations of flows past a sphere are chosen as a benchmark representative of a class of engineering flows past obstacles. The methodology is further extended to moderate Reynolds number, stably stratified flows under gravity, for Froude numbers that typify the characteristic regimes of natural flows past distinct isolated features of topography in weather and climate models. The key elements of the proposed method consist of the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) and a robust non-symmetric Krylov-subspace elliptic solver. The solutions employ a finite volume spatial discretisation on unstructured and hybrid meshes and benefit from a collocated arrangement of all flow variables while being inherently stable. The development includes the implementation of viscous terms with the detached-eddy simulation (DES) approach employed for turbulent flows. Results demonstrate that the proposed methodology enables direct comparisons of the numerical solutions with corresponding laboratory studies of viscous and stratified flows while illustrating accuracy, robustness and flexibility of the NFT schemes. The presented simulations also offer a better insight into stably stratified flows past a sphere.

FVM 1.0: a nonhydrostatic finite-volume dynamical core for the IFS

Abstract. We present a nonhydrostatic finite-volume global atmospheric model formulation for numerical weather prediction with the Integrated Forecasting System (IFS) at ECMWF and compare it to the established operational spectral-transform formulation. The novel Finite-Volume Module of the IFS (henceforth IFS-FVM) integrates the fully compressible equations using semi-implicit time stepping and non-oscillatory forward-in-time (NFT) Eulerian advection, whereas the spectral-transform IFS solves the hydrostatic primitive equations (optionally the fully compressible equations) using a semi-implicit semi-Lagrangian scheme. The IFS-FVM complements the spectral-transform counterpart by means of the finite-volume discretization with a local low-volume communication footprint, fully conservative and monotone advective transport, all-scale deep-atmosphere fully compressible equations in a generalized height-based vertical coordinate, and flexible horizontal meshes. Nevertheless, both the finite-volume and spectral-transform formulations can share the same quasi-uniform horizontal grid with co-located arrangement of variables, geospherical longitude–latitude coordinates, and physics parameterizations, thereby facilitating their comparison, coexistence, and combination in the IFS. We highlight the advanced semi-implicit NFT finite-volume integration of the fully compressible equations of IFS-FVM considering comprehensive moist-precipitating dynamics with coupling to the IFS cloud parameterization by means of a generic interface. These developments – including a new horizontal–vertical split NFT MPDATA advective transport scheme, variable time stepping, effective preconditioning of the elliptic Helmholtz solver in the semi-implicit scheme, and a computationally efficient implementation of the median-dual finite-volume approach – provide a basis for the efficacy of IFS-FVM and its application in global numerical weather prediction. Here, numerical experiments focus on relevant dry and moist-precipitating baroclinic instability at various resolutions. We show that the presented semi-implicit NFT finite-volume integration scheme on co-located meshes of IFS-FVM can provide highly competitive solution quality and computational performance to the proven semi-implicit semi-Lagrangian integration scheme of the spectral-transform IFS.

Extension of a Non-Oscillatory Forward-in-Time Scheme to Detached Eddy Simulation of Stratified Turbulent Flows With Unstructured Meshes

A detached eddy simulation algorithm is developed based on the non-oscillatory forward-in-time (NFT) multidimensional positive definite advection transport algorithm (MPDATA) scheme for stratified turbulent flows with low or moderate Reynolds numbers on unstructured meshes. NFT MPDATA is extended to viscous turbulent flows for the first time. Viscous diffusion terms and an explicit sub-grid-scale model are included. The working viscosity of the modified Spalart-Allmaras one equation model is transported by an algorithm based on the NFT MPDATA template. The scheme is semi-implicit which solves the linear equation system once per time step. The advantage of the scheme is efficiency and second-order accuracy. Furthermore, an edge-based data structure is adopted such that our scheme is suitable for both structured and unstructured meshes. The linear equation system is solved by an efficient algebraic multigrid (AMG) method. The algorithm is validated using test cases with Reynolds numbers 200 and 5000 over a range of Froude numbers. The flow patterns and dynamic drag coefficients are analysed and compared to numerical and experimental data in literature.

Implementation of a non-hydrostatic, adaptive-grid dynamics core in CAM3. Part I: comparison of dynamics cores in aqua-planet simulations

We report on our implementation of EULAG as a dynamical core in the Community Atmospheric Model (CAM). EULAG is a non-hydrostatic, parallel computational model for all-scale geophysical flows. EULAG’s name derives from its two computational options: EULerian (flux form) or semi-LAGrangian (advective form). The model combines nonoscillatory forward-in-time (NFT) numerical algorithms with a robust elliptic Krylov solver. A signature feature of EULAG is that it is formulated in generalized coordinates. In particular, this enables grid adaptivity. In total, these features give EULAG novel advantages over the existing dynamical cores in CAM. This paper uses a series of aqua-planet simulations to demonstrate that CAM-EULAG results compare favorably with those from CAM simulations at standard CAM resolution that use current finite volume or Eulerian-spectral dynamical core options. We also show that the grid adaptivity implemented in CAM3-EULAG allows higher resolution in selected regions without causing anomalous behavior such as spurious wave reflection.

Building resolving large-eddy simulations and comparison with wind tunnel experiments

We perform large-eddy simulations (LES) of the flow past a scale model of a complex building. Calculations are accomplished using two different methods to represent the edifice. The first method employs the standard Gal-Chen and Somerville terrain-following coordinate transformation, common in mesoscale atmospheric simulations. The second method uses an immersed boundary approach, in which fictitious body forces in the equations of motion are used to represent the building by attenuating the flow to stagnation within a time comparable to the time step of the model. Both methods are implemented in the same hydrodynamical code (EULAG) using the same nonoscillatory forward-in-time (NFT) incompressible flow solver based on the multidimensional positive definite advection transport algorithms (MPDATA). The two solution methods are compared to wind tunnel data collected for neutral stratification. Profiles of the first- and second-order moments at various locations around the model building show good agreement with the wind tunnel data. Although both methods appear to be viable tools for LES of urban flows, the immersed boundary approach is computationally more efficient. The results of these simulations demonstrate that, contrary to popular opinion, continuous mappings such as the Gal-Chen and Somerville transformation are not inherently limited to gentle slopes. Calculations for a strongly stratified case are also presented to point out the substantial differences from the neutral boundary layer flows.

MPDATA error estimator for mesh adaptivity

AbstractIn multidimensional positive definite advection transport algorithm (MPDATA) the leading error as well as the first‐ and second‐order solutions are known explicitly by design. This property is employed to construct refinement indicators for mesh adaptivity. Recent progress with the edge‐based formulation of MPDATA facilitates the use of the method in an unstructured‐mesh environment. In particular, the edge‐based data structure allows for flow solvers to operate on arbitrary hybrid meshes, thereby lending itself to implementations of various mesh adaptivity techniques. A novel unstructured‐mesh nonoscillatory forward‐in‐time (NFT) solver for compressible Euler equations is used to illustrate the benefits of adaptive remeshing as well as mesh movement and enrichment for the efficacy of MPDATA‐based flow solvers. Validation against benchmark test cases demonstrates robustness and accuracy of the approach. Copyright © 2005 John Wiley & Sons, Ltd.

Laboratory for internal gravity‐wave dynamics: the numerical equivalent to the quasi‐biennial oscillation (QBO) analogue

AbstractWe have extended the classical terrain‐following coordinate transformation of Gal–Chen and Somerville to a broad class of time‐dependent vertical domains. The proposed extension facilitates modelling of undulating vertical boundaries in various areas of computational fluid dynamics. The theoretical development and the efficient numerical implementation have been documented in the context of the generic Eulerian/semi‐Lagrangian, non‐oscillatory forward in time (NFT), nonhydrostatic model framework. In particular, it allows the simulation of stratified flows with intricate geometric, time‐dependent boundary forcings, either at the top or at the bottom of the domain. We have applied our modelling framework in the direct numerical simulation of the celebrated laboratory experiment of Plumb and McEwan creating the numerical equivalent to their quasi‐biennial oscillation (QBO) analogue. The QBO represents a conspicuous example of a fundamental dynamical mechanism with challenging detail, which is difficult to deduce from experimental evidence alone. A series of 2D and 3D simulations demonstrate the ability to reproduce the laboratory results. The numerical experiments identify the developing periodically reversing mean flow pattern primarily as a wave–wave mean flow interaction‐driven phenomenon. The results not only enhance the confidence in the numerical approach but further elevate the importance of the laboratory setup in its fundamental similarity to the atmosphere, while allowing the study of the principal atmospheric mechanisms and their numerical realizability in a confined ‘laboratory’ environment. Copyright © 2005 John Wiley & Sons, Ltd.

An all-scale anelastic model for geophysical flows: dynamic grid deformation

We have developed an adaptive grid-refinement approach for simulating geophysical flows on scales from micro to planetary. Our model is nonoscillatory forward-in-time (NFT), nonhydrostatic, and anelastic. The major focus in this effort to date has been the design of a generalized mathematical framework for the implementation of deformable coordinates and its efficient numerical coding in a generic Eulerian/semi-Lagrangian NFT format. The key prerequisite of the adaptive grid is a time-dependent coordinate transformation, implemented rigorously throughout the governing equations of the model. The transformation enables mesh refinement indirectly via dynamic change of the metric coefficients, while retaining advantages of Cartesian mesh calculations (speed, low memory requirements, and accuracy) conducted fully in the computational domain. Diverse test results presented in this paper – simulations of a traveling stratospheric inertio-gravity-wave packet (with numerically advected dense-mesh region) and an idealized climate of the Earth (with analytically prescribed adaptive transformations) – demonstrate the potential and the efficacy of the new deformable grid model for tracing targeted flow features and dynamically adjusting to prescribed undulations of model boundaries.