Strong Conservation Form Research Articles

Advanced modelling of lubricated interfaces in general curvilinear grids

Tackling fluid-flow problems involving intricate surface geometries has been the catalyst for a plethora of numerical investigations aimed at accommodating curved complex boundaries. An example is the application of body-fitted curvilinear coordinate transformation, where the one-to-one correspondence of grid points from a non-orthogonal curvilinear grid in the physical domain to an orthogonal grid in the computational domain is achieved. In lubricated interfaces, such conversion is challenging due to the complexity of the governing equations in the mapped-grid, the numerical instabilities exhibited by their non-linearities and the severity of the operating conditions. The present contribution proposes a Reynolds-based, finite volume fluid–structure interaction (FSI) framework for solving thermal elastohydrodynamic lubrication (TEHL) problems mapped onto orthogonal grids in the computational domain. We demonstrate how the strong conservation form of the pertinent governing equations can be expressed in three-dimensional curvilinear grids and discretised using the finite volume method to ensure fluid-flow conservation and enforce mass-conserving cavitation conditions. Numerical and experimental benchmarks showcase the robustness and versatility of the proposed framework to simulate a diverse range of lubrication problems, hence achieving a predictive computational tool that would enable a shift towards tribology-aware design.

A meshfree radial basis function method for simulation of multi‐dimensional conservation problems

AbstractMany computational fluid dynamics problems utilize finite volume frameworks for simulation, due to the simplifications provided by conservative formulation of the driving partial differential equations (PDEs). However, fluid dynamics applications can often involve temporal shifts in the domain structure—such as moving boundaries, or pore structure changes—requiring mesh adaptation throughout computation. These mesh adaptations often render classical numerical methods such as the finite volume method infeasible, due to their reliance on a well‐defined static mesh structure. This limitation has led to the development of a wide variety of meshless methods—techniques that can simulate PDEs without requiring a rigid connective structure between nodes. However, most meshless methods are typically based on finite element or finite difference formulations, and the limited number of meshless finite volume methods (MFVMs) either introduce a weak background mesh, or use weak‐form approximations that do not take full advantage of the strong conservative form of the driving equations. Addressing this gap within this study we outline a meshfree numerical scheme for simulation of partial different equations, based on strong‐form finite volume style formulations. Building upon the previously developed MFVM, this technique uses radial basis functions to interpolate the problem domain, and approximate fluxes in a disjoint finite volume scheme, removing reliance on a mesh structure. We present method derivation, including promising new techniques for enforcing boundary conditions in a meshless environment. Following this we discuss method accuracy and computational performance across a variety of problems in two and three dimensions. We then illustrate how this method may prove beneficial for applications in porous media modeling, and computational fluid dynamics. For completeness, we provide a sensitivity analysis of the method hyper‐parameters and investigate the conservative properties of the method. We also illustrate similarities of this approach to the widely used meshless point collocation methods. We close with a discussion of the strengths, limitations, and broader applicability of the technique.

A horizontally explicit, vertically implicit (HEVI) discontinuous Galerkin scheme for the 2-dimensional Euler and Navier-Stokes equations using terrain-following coordinates

A solver for the Euler equations with optional diffusion based on the Discontinuous Galerkin (DG) method is presented. To be usable for simulation problems in the atmosphere from the global scale down to the meso-scale, the horizontally explicit, vertically implicit (HEVI) approach is applied to the DG discretization, to avoid tiny time-steps by thin grid cells. To consider orography in the correct approximation order, terrain-following coordinates are used, and to nevertheless keep the local conservation properties of DG, the equations are formulated in the strong conservation form. IMEX-Runge-Kutta time integration enables at least third order approximation in time. In a similar manner, diffusion both as a physical process (for turbulence parameterization) and as a physically motivated stabilization mechanism is treated by the Bassi, Rebay (1997) approach in terrain-following coordinates and in a conserving manner, too. Several test cases relevant for the atmosphere demonstrate the validity of the approach. Although most of the presented theory is applicable both in two and three dimensions (2D and 3D), these test cases are purely 2D. They show, in particular, that the proposed scheme seems to tolerate very steep terrain.

Numerical study of shock-disturbances interaction in hypersonic inviscid flows with real gas effects using high-order WENO scheme

In the present study, the shock-disturbances interaction in hypersonic inviscid flows with real gas effects is studied by applying a high-order accurate numerical method with the shock capturing technique. To consider real gas effects, the equilibrium air model is utilized here. The strong conservative form of the unsteady compressible Euler equations in the 2D generalized curvilinear coordinates is formulated and the resulting system of equations for the equilibrium air model is discretized by using the fifth-order finite-difference WENO scheme in space and the explicit third-order TVD Runge–Kutta scheme in time to provide a highly accurate and robust equilibrium airflow solver. The solution methodology adopted based on the high-order WENO scheme with the shock capturing technique can reasonably simulate the strong discontinuities in the solution domain without needing any numerical artifact and this feature makes the solution method suitable for the shock-disturbances interaction in hypersonic flows in which real gas effects are to be considered. To study the physical phenomenon of the shock-disturbances interaction problem in high-speed inviscid equilibrium airflows, two test cases are simulated by applying the solution methodology adopted. In the first test case, the interaction between the incoming fast acoustic wave with the bow shock formed around a cylinder with the nose radius of RN=0.0381 m at M∞=8.03 and T∞=182.333 K is simulated. In the second test case, the cylinder radius is taken RN=2.54×10−2 m and the upstream velocity and temperature are u∞=5590m/s and T∞=1833 K, respectively, and all kinds of free stream disturbances, including the entropy wave and the fast and slow acoustic waves, for different wave numbers are considered. The computations are carried out for both the perfect gas and equilibrium air models and the effects of real gas on both the mean and disturbances fields are also studied. The present study introduces the first-known numerical investigation of the shock-disturbances interaction in hypersonic flow over blunt noses with real gas effects in which the numerical results obtained are thoroughly validated by the analytical/theoretical ones and good agreement is found.

Large-scale heating effects on global parameters for flow past a square cylinder at different cylinder inclinations

The combined effects of heating and cylinder inclination on mixed convective flow past a square cylinder subjected to high heating is investigated numerically using a non-Oberbeck-Boussinesq model. A two-dimensional, unsteady, laminar, viscous and compressible flow model is utilized. In order to account for the large-scale heating effects; variations in transport and thermo-physical properties are considered. The flow parameters such as Reynolds number (Re), Prandtl number (Pr), Mach number (M), Froude number (Fr) and free-stream inclination (α) are kept fixed as Re=100,Pr=0.71,M=0.1,Fr=1.0 and α=0o, respectively. The cylinder inclination (ϕ) is varied from 0o to 45o in a step of 10o, while heating levels (ϵ) defined as (Tw−T∞)/T∞ is varied from 0 to 1, in a step of 0.2. The governing equations, in strong conservative form, transformed into body fitted curvilinear co-ordinates, are solved numerically using particle velocity upwind (PVU-M+) scheme employing finite difference method. The effects of cylinder inclination and heating on global parameters such as drag coefficient, lift coefficient, moment coefficient, Strouhal number and Nusselt number are presented. It is observed that for any heating level, mean drag coefficient increases with increase in cylinder inclination. Also at fixed ϕ, mean drag coefficient increases with heating. It is found that in steady flow regime increase in ϕ, decreases the heat transfer rate. In unsteady flow regime, increase in ϕ, increases the heat transfer rate. It is also observed that at fixed cylinder inclination mean Nusselt number increases with increase in heating level.

Numerical simulation of shock-disturbances interaction in high-speed compressible inviscid flow over a blunt nose using weighted essentially non-oscillatory scheme

In the present study, shock-disturbances interaction in high-speed compressible inviscid flow is simulated utilizing the weighted essentially non-oscillatory (WENO) scheme by employing the shock-capturing technique. For this aim, the two-dimensional Euler equations in strong conservative form are discretized by using the explicit third-order TVD Runge–Kutta scheme in time and the fifth-order WENO finite difference scheme in space. The main advantage of using the WENO scheme is its capability for properly solving the discontinuities in the domain without needing any artificial viscosity, limiter function or filter. Hence, this scheme is stable, and thus, it is suitable for simulating very high-speed flows with strong shocks. Three numerical tests are presented to demonstrate the remarkable capability of the WENO scheme and the corresponding numerical results are compared with the available analytical and numerical studies in the literature. The final and main test case is the steady and unsteady numerical simulation of the high-speed compressible inviscid flow past a two-dimensional circular cylinder. For this test case, firstly, the steady numerical solution at Mach=8.03 is provided. Then, the disturbances field is added to the upstream flow field and the unsteady numerical solution is performed for the fast and slow acoustic and also entropy waves. A sensitivity study is also performed to check the effects of the grid size and the type of flux splitting method on the solution. Both the steady and unsteady numerical solutions are compared with the existing analytical and numerical solutions and some conclusions are made.

Conservative high-order flux-reconstruction schemes on moving and deforming grids

Abstract An appropriate procedure to construct symmetric conservative metrics is presented for the high-order conservative flux-reconstruction scheme on three-dimensionally moving and deforming grids. The present framework enables direct discretization of the strong conservation form of governing equations without any errors in the freestream preservation and global conservation properties. We demonstrate that a straightforward implementation of the symmetric conservative metrics often fails to construct metric polynomials having the same order as a solution polynomial, which severely degrades the numerical accuracy. On the other hand, the symmetric conservative metrics constructed using an appropriate procedure can preserve the freestream solution regardless of the order of shape functions. Moreover, a convecting vortex is more accurately computed on deforming grids. The global conservation property is also demonstrated numerically for the convecting vortex on deforming grids.

Artificial viscosity in comoving curvilinear coordinates: towards a differential geometrically consistent implicit advection scheme

AbstractWe propose a modification for the tensor of artificial viscosity employable for generally comoving, curvilinear grids. We present a strong conservation form for the equations of radiation hydrodynamics for studying nonlinear pulsations of stars. However, the modification we propose is of general mathematical nature. We study a differential geometrically consistent artificial viscosity analytically and visualize a comparison of our approach to previous implementations by applying it to a simple self-similar velocity field which has a direct application in stars as the fundamental mode of pulsation is radial. We first give a general introduction to artificial viscosity and motivate its application in numerical computations. We then show how a tensor of artificial viscosity has to be designed when going beyond common static Eulerian or Lagrangian comoving rectangular grids. We derive and state the modified equations which include metrical terms that adjust the isotropic (pressure) part of the tensor of artificial viscosity.

A strong conservative Riemann solver for the solution of the coupled Maxwell and Navier–Stokes equations

The coupled system of the Navier–Stokes and Maxwell equations are recast into a strong conservative form, which allows the fluid coupling to the Maxwell system to be written in terms of flux divergence rather than explicit source terms. This effectively removes source terms from the Navier–Stokes equations, although retaining an exact coupling to the electromagnetics. While this relieves the stiff source terms and potentially stabilizes the system, it introduces a much more complicated eigenstructure to the governing equations. The flux Jacobian and eigenvectors for this strong conservative system are presented in the current paper for the first time. An approximate Riemann solver based upon these eigenvectors is then introduced and tested. The solver is implemented in a preconditioned, dual-time implicit form. Validations for classic one- and two-dimensional problems are presented, and the performances of the new formulation and the traditional source-coupled formulation are compared.

A Mach‐uniform preconditioner for incompressible and subsonic flows

SUMMARYIn this study, a novel Mach‐uniform preconditioning method is developed for the solution of Euler equations at low subsonic and incompressible flow conditions. In contrast to the methods developed earlier in which the conservation of mass equation is preconditioned, in the present method, the conservation of energy equation is preconditioned, which enforces the divergence free constraint on the velocity field even at the limiting case of incompressible, zero Mach number flows. Despite most preconditioners, the proposed Mach‐uniform preconditioning method does not have a singularity point at zero Mach number. The preconditioned system of equations preserves the strong conservation form of Euler equations for compressible flows and recovers the artificial compressibility equations in the case of zero Mach number. A two‐dimensional Euler solver is developed for validation and performance evaluation of the present formulation for a wide range of Mach number flows. The validation cases studied show the convergence acceleration, stability, and accuracy of the present Mach‐uniform preconditioner in comparison to the non‐preconditioned compressible flow solutions. The convergence acceleration obtained with the present formulation is similar to those of the well‐known preconditioned system of equations for low subsonic flows and to those of the artificial compressibility method for incompressible flows. Copyright © 2013 John Wiley & Sons, Ltd.

A high‐resolution scheme for compressible multicomponent flows with shock waves

AbstractA simple methodology for a high‐resolution scheme to be applied to compressible multicomponent flows with shock waves is investigated. The method is intended for use with direct numerical simulation or large eddy simulation of compressible multicomponent flows. The method dynamically adds non‐linear artificial diffusivity locally in space to capture different types of discontinuities such as a shock wave, contact surface or material interface while a high‐order compact differencing scheme resolves a broad range of scales in flows. The method is successfully applied to several one‐dimensional and two‐dimensional compressible multicomponent flow problems with shock waves. The results are in good agreement with experiments and earlier computations qualitatively and quantitatively. The method captures unsteady shock and material discontinuities without significant spurious oscillations if initial start‐up errors are properly avoided. Comparisons between the present numerical scheme and high‐order weighted essentially non‐oscillatory (WENO) schemes illustrate the advantage of the present method for resolving a broad range of scales of turbulence while capturing shock waves and material interfaces. Also the present method is expected to require less computational cost than popular high‐order upwind‐biased schemes such as WENO schemes. The mass conservation for each species is satisfied due to the strong conservation form of governing equations employed in the method. Copyright © 2010 John Wiley & Sons, Ltd.

A New Spatial Discretization Strategy of the Convective Flux Term for the Hyperbolic Conservation Laws

:In this work, a new spatial discretization scheme for flows governed by the hyperbolic conservation laws is proposed. The spatial discretization involves the concept of classical particle velocity upwinding (PVU) for the convective flux term in the hyperbolic conservation laws. The novelty of the approach lies in the use of the fluid particle velocity or the entropy wave speed at the cell interface to ascertain the upwind direction. The cell face convective fluxes are obtained from a first order or a second order upwind biased interpolation, depending on whether the cell under consideration lies in the vicinity of a discontinuity or in a region of steep gradients in the solution. The discontinuities or regions of steep gradients are detected by employing a smoothness indicator function as employed in some of the earlier studies. The proposed spatial discretization strategy has been combined with a two step, second order explicit time integration strategy for the application to the solution of the unsteady Euler/Navier-Stokes equations in the strong conservation form. Test cases involving two 1-D Riemann problems, three 2-D inviscid supersonic flow problems and a 2-D viscous supersonic flow problem, have been employed to establish the validity of the procedure and to assess the performance of the proposed strategy. The proposed PVU scheme performs quite favorably in comparison to conventional schemes. From the point of view of implementation, particularly in multidimensional scenarios, this strategy offers a good balance of accuracy and simplicity.

New Dissipative Leapfrog Finite Difference Scheme for Elastodynamic Simulation in Noninertial Frames

A new approach to numerically solving problems of the elastodynamics of arbitrarily shaped bodies with nonuniform material properties in rotation has been proposed in the present study. The present approach is based on first principles and is well suited to the study of problems of fluid-structure interaction and wave-based structural health monitoring. The governing hyperbolic partial differential equations of elastodynamics are cast in strong conservation form in the noninertial frame of reference and in generalized curvilinear coordinates. The results are obtained directly in the time domain. No special boundary treatment is required at material interfaces, and predictions can be made with and without material damage in a given structure for damage detection of the structure. The effect on the dynamics of a rotating annulus due to centrifugal force and, additionally, Coriolis force at high rates of rotation is studied. The partial differential equations are solved using a new dissipative time-centered leapfrog scheme presented here. A one-dimensional theoretical analysis of the numerical scheme shows that the new scheme has well-defined stability and dissipation characteristics. A comparison of the new scheme with a conventional dissipative time-centered leapfrog scheme shows that much larger time steps can be taken with the new scheme than allowed by the existing scheme. A dissipation-dependence study demonstrates that as the dissipation parameter is decreased, the steady-state solution uniformly approaches the exact solution. For larger values of this parameter, although the natural frequency of the system is not altered, the system vibrations damp too quickly, and the steady-state solution deviates from the exact steady-state solution. The larger the dissipation parameter, the larger this deviation from the exact steady state. The present scheme is very simple to incorporate in a finite difference simulation of elastodynamics in generalized curvilinear coordinates, and it has been used in the software FiDDLE, which solves the velocity-stress system of elastodynamic partial differential equations. It will be used in simulating fluid-elastodynamic interaction of the Mars Science Laboratory canopies and other decelerators such as inflatable aerodynamic devices as well as in wave-based structural health monitoring of these systems. Other applications include elastodynamics of turbomachinery blades, computer disk drive, and cutting-tool industry applications.

Conservation laws in curvilinear coordinates: A short proof of Vinokur’s Theorem using differential forms

In computational fluid dynamics it is important to maintain strong conservation form in the equations of motion regardless of the coordinate system used. Vinokur’s Theorem says that conservation form can always be maintained. However, the proof is long and has non-obvious steps. In this note a short proof of Vinokur’s Theorem is given which is both simple and illuminating. It uses the theory of differential forms which may also be useful in other algorithmic constructions.

High-order triangle-based discontinuous Galerkin methods for hyperbolic equations on a rotating sphere

High-order triangle-based discontinuous Galerkin (DG) methods for hyperbolic equations on a rotating sphere are presented. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. The finite element-type area integrals are evaluated using order 2 N Gauss cubature rules. This leads to a full mass matrix which, unlike for continuous Galerkin (CG) methods such as the spectral element (SE) method presented in Giraldo and Warburton [A nodal triangle-based spectral element method for the shallow water equations on the sphere, J. Comput. Phys. 207 (2005) 129–150], is small, local and efficient to invert. Two types of finite volume-type flux integrals are studied: a set based on Gauss–Lobatto quadrature points (order 2 N − 1) and a set based on Gauss quadrature points (order 2 N). Furthermore, we explore conservation and advection forms as well as strong and weak forms. Seven test cases are used to compare the different methods including some with scale contractions and shock waves. All three strong forms performed extremely well with the strong conservation form with 2 N integration being the most accurate of the four DG methods studied. The strong advection form with 2 N integration performed extremely well even for flows with shock waves. The strong conservation form with 2 N − 1 integration yielded results almost as good as those with 2 N while being less expensive. All the DG methods performed better than the SE method for almost all the test cases, especially for those with strong discontinuities. Finally, the DG methods required less computing time than the SE method due to the local nature of the mass matrix.

Hybrid Compressible-Incompressible Numerical Method for Transient Drop-Gas Flows

A numerical method for drops in which the liquid phase is modeled as an incompressible fluid while the gas phase is modeled as a compressible fluid is presented. An unstructured grid that conforms to the deforming interface at all instants in time is employed. Finite volume discretization of the strong conservative form of the governing equations is solved with an implicit iterative procedure that simultaneously solves governing equations for the two phases and the boundary conditions at the interface. The numerical accuracy of the code is assessed by comparing the computed transient drag of a decelerating solid sphere with prior computed results published in the literature and comparing the computed behavior of an oscillating drop with the classical solution of the problem. The numerical method is assessed in axisymmetric computations of deforming and decelerating liquid drops. The transient drag of the drop is compared with computed results for a solid sphere.

Role of Artificial Dissipation Scaling and Multigrid Acceleration in Numerical Solutions of the Depth-Averaged Free-Surface Flow Equations

A numerical model is developed for solving the depth-averaged, open-channel flow equations in generalized curvilinear coordinates. The equations are discretized in space in strong conservation form using a space-centered, second-order accurate finite-volume method. A nonlinear blend of first- and third-order accurate artificial dissipation terms is introduced into the discrete equations to accurately model all flow regimes. Scalar- and matrix-valued scaling of the artificial dissipation terms are considered and their effect on the accuracy of the solutions is evaluated. The discrete equations are integrated in time using a four-stage explicit Runge–Kutta method. For the steady-state computations, local time stepping, implicit residual smoothing, and multigrid acceleration are used to enhance the efficiency of the scheme. The numerical model is validated by applying it to calculate steady and unsteady open-channel flows. Extensive grid sensitivity studies are carried out and the potential of multigrid acce...

Extension of a Compressible Code Toward the Incompressible Limit

Pressure is used as a dependent variable for consistent extension toward incompressible flows. The proposed method exploits the fact that the divergence-free constraint on the velocity field for incompressible flow derives from the energy equation. In the compressible regime the implicit elliptic pressure equation transforms into an explicit hyperbolic equation, leading to the common formulation used in explicit compressible codes. The method strictly respects the strong conservation form, and reliable shock capturing is established by the use of a proven approximate Riemann solver for flux computation

Proper Orthogonal Decomposition of Turbine Rotor-Stator Interaction

Numerical simulations of the flow in a one-stage turbine are carried out to investigate the feasibility of constructing a reduced-order model via Galerkin methods. The flow in the turbine is modeled by the unsteady Reynolds-averaged Navier-Stokes equations. The governing equations are written in the strong conservation form and solved using a fully implicit, finite difference approximation. By the use of the proper orthogonal decomposition (POD), spatial dominant features, also known as POD modes or eigenfunctions, are identified and separated from the spatiotemporal dynamics of the turbine flow. The POD reconstructed solutions indicate that a significant portion of the original dynamics is captured by a few modes. The solution reconstructed using the first 40 modes captures more than 99% of the energy spectrum, whereas the error of the energy variable is less than 0.6%, and the error of skin friction is less than 1.5%. Phase-space plots further indicate the existence of low-dimensional dynamics, which supports the validity of a reduced-order model for turbine flow.

Numerical Simulation of Combustion and Rotor‐Stator Interaction in a Turbine Combustor

This article presents the development of a numerical algorithm for the computation of flow and combustion in a turbine combustor. The flow and combustion are modeled by the Reynolds‐averaged Navier‐Stokes equations coupled with the species‐conservation equations. The chemistry model used herein is a two‐step, global, finite‐rate combustion model for methane and combustion gases. The governing equations are written in the strong conservation form and solved using a fully implicit, finite‐difference approximation. The gas dynamics and chemistry equations are fully decoupled. A correction technique has been developed to enforce the conservation of mass fractions. The numerical algorithm developed herein has been used to investigate the flow and combustion in a one‐stage turbine combustor.