Inviscid Problem Research Articles

A linearity preserving nodal variation limiting algorithm for continuous Galerkin discretization of ideal MHD equations

In this work, a stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm. This involves adding some artificial diffusion to the high order, semi-discrete method and mass lumping in the time derivative term. The result is a low order method that provides local extremum diminishing properties for hyperbolic systems. The difference between the low order method and the high order method is scaled element-wise using a limiter and added to the low order scheme. The limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy. The stabilization also involves an optional consistent background high order dissipation that reduces phase errors. The resulting stabilized scheme is a semi-discrete method that can be applied to inviscid shock MHD problems and may be even extended to resistive and viscous MHD problems. To satisfy the divergence free constraint of the MHD equations, we add parabolic divergence cleaning to the system. Various time integration methods can be used to discretize the scheme in time. We demonstrate the robustness of the scheme by solving several shock MHD problems.

On energy exchanges between eddies and the mean flow in quasigeostrophic turbulence

We study the term in the eddy energy budget of continuously stratified quasigeostrophic turbulence that is responsible for energy extraction by eddies from the background mean flow. This term is a quadratic form, and we derive Euler–Lagrange equations describing its eigenfunctions and eigenvalues, the former being orthogonal in the energy inner product and the latter being real. The eigenvalues correspond to the instantaneous energy growth rate of the associated eigenfunction. We find analytical solutions in the Eady problem. We formulate a spectral method for computing eigenfunctions and eigenvalues, and compute solutions in Phillips-type and Charney-type problems. In all problems, instantaneous growth is possible at all horizontal scales in both inviscid problems and in problems with linear Ekman friction. We conjecture that transient growth at small scales is matched by linear transfer to decaying modes with the same horizontal structure, and we provide simulations supporting the plausibility of this hypothesis. In Charney-type problems, where the linear problem has exponentially growing modes at small scales, we expect net energy extraction from the mean flow to be unavoidable, with an associated nonlinear transfer of energy to dissipation.

Central upwind scheme based immersed boundary method for compressible flows around complex geometries

The present work combines easy to implement high resolution central upwind scheme (CUS) with a ghost cell based discrete forcing immersed boundary method (IBM) and proposes a novel CUS-IBM finite volume solver to simulate high-speed compressible inviscid/viscous flows around complex geometries on Cartesian grids. Proposed CUS-IBM solver has been tested for a variety of inviscid and viscous compressible flow problems, ranging from transonic to supersonic flow speeds (0.625 < M < 3), involving flow over different geometries, i.e., bump, airfoil and cylinder. Results from our computations clearly show that solutions computed with the proposed method are in good agreement with the numerical data available in literature for the various inviscid/viscous and external/internal compressible flow problems. The present study demonstrates that the proposed CUS-IBM exhibits nominal second-order accuracy in space and is an easy to implement, robust and promising alternative to the methods involving Riemann solver and characteristics decomposition.

A face-area-weighted ‘centroid’ formula for finite-volume method that improves skewness and convergence on triangular grids

This paper proposes a face-area-weighted ‘centroid’ as a superior alternative to the geometric centroid for defining a local origin in a cell-centered finite-volume method on triangular grids. It is demonstrated theoretically and numerically that the face-area-weighted ‘centroid’ can reduce grid skewness and improve iterative convergence for triangular grids. It is also shown that source terms do not have to be integrated over a cell and can be evaluated simply at the local origin without losing the design order of accuracy. Numerical results demonstrate that the face-area-weighted ‘centroid’ improves iterative convergence of an implicit defect-correction second-order finite-volume solver for inviscid and viscous flow problems on regular and irregular triangular grids.

M6 USING PRS SCORES TO CLARIFY NOSOLOGICAL STRUCTURE OF ADHD AND OVERLAPPING EMOTION-RELATED TRAIT MEASURES

In this work, a stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm. This involves adding some artificial diffusion to the high order, semi-discrete method and mass lumping in the time derivative term. The result is a low order method that provides local extremum diminishing properties for hyperbolic systems. The difference between the low order method and the high order method is scaled element-wise using a limiter and added to the low order scheme. The limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy. The stabilization also involves an optional consistent background high order dissipation that reduces phase errors. The resulting stabilized scheme is a semi-discrete method that can be applied to inviscid shock MHD problems and may be even extended to resistive and viscous MHD problems. To satisfy the divergence free constraint of the MHD equations, we add parabolic divergence cleaning to the system. Various time integration methods can be used to discretize the scheme in time. We demonstrate the robustness of the scheme by solving several shock MHD problems.

Relinearization of the error transport equations for arbitrarily high-order error estimates

A higher-order accurate discretization error estimation procedure for finite-volume schemes is presented. Discretization error estimates are computed using the linearized error transport equations (ETE). ETE error estimates are applied as a correction to the primal solution. The ETE are then relinearized about the corrected primal solution, and discretization error estimates are recomputed. This process, referred to as ETE relinearization, is performed in an iterative manner to successively increase the accuracy of discretization error estimates. Under certain conditions, ETE relinearization is shown to correct error estimates, or equivalently the entire primal solution, to higher-order accuracy. In terms of computational cost, ETE relinearization has a competitive advantage over conventional higher-order discretizations when used as a form of defect correction for the primal solution. Furthermore, ETE relinearization is shown to be particularly useful for problems where the error incurred by the linearization of the ETE cannot be neglected. Results are presented for several steady-state inviscid and viscous flow problems using both structured and unstructured meshes.

Self-channelisation and levee formation in monodisperse granular flows

Dense granular flows can spontaneously self-channelise by forming a pair of parallel-sided static levees on either side of a central flowing channel. This process prevents lateral spreading and maintains the flow thickness, and hence mobility, enabling the grains to run out considerably further than a spreading flow on shallow slopes. Since levees commonly form in hazardous geophysical mass flows, such as snow avalanches, debris flows, lahars and pyroclastic flows, this has important implications for risk management in mountainous and volcanic regions. In this paper an avalanche model that incorporates frictional hysteresis, as well as depth-averaged viscous terms derived from the $\unicode[STIX]{x1D707}(I)$-rheology, is used to quantitatively model self-channelisation and levee formation. The viscous terms are crucial for determining a smoothly varying steady-state velocity profile across the flowing channel, which has the important property that it does not exert any shear stresses at the levee–channel interfaces. For a fixed mass flux, the resulting boundary value problem for the velocity profile also uniquely determines the width and height of the channel, and the predictions are in very good agreement with existing experimental data for both spherical and angular particles. It is also shown that in the absence of viscous (second-order gradient) terms, the problem degenerates, to produce plug flow in the channel with two frictionless contact discontinuities at the levee–channel margins. Such solutions are not observed in experiments. Moreover, the steady-state inviscid problem lacks a thickness or width selection mechanism and consequently there is no unique solution. The viscous theory is therefore a significant step forward. Fully time-dependent numerical simulations to the viscous model are able to quantitatively capture the process in which the flow self-channelises and show how the levees are initially emplaced behind the flow head. Both experiments and numerical simulations show that the height and width of the channel are not necessarily fixed by these initial values, but respond to changes in the supplied mass flux, allowing narrowing and widening of the channel long after the initial front has passed by. In addition, below a critical mass flux the steady-state solutions become unstable and time-dependent numerical simulations are able to capture the transition to periodic erosion–deposition waves observed in experiments.

On the expanding configurations of viscous radiation gaseous stars: the isentropic model

In this work, we study the stability of the expanding configurations for radiation gaseous stars. Such expanding configurations exist for the isentropic model. In fact, they are given as a class of self-similar solutions to the associated dynamic system, which can be viewed as the asymptote for the system with viscosity coefficients satisfying for the monatomic gas as . Compared with previous works on the inviscid problem (see Hadžić and Jang (2017 Commun. Pure Appl. Math. LXXI 827–91)), we focus on the effect of the viscosity on the expanding configurations. With respect to small perturbations, it is shown that the self-similarly expanding homogeneous solutions of the isentropic model are unstable, while the linearly expanding homogeneous solutions of the isentropic model are stable. The corresponding problem for inviscid flows has been studied in Hadžić and Jang (2017 Commun. Pure Appl. Math. LXXI 827–91).

Note on eigenvalue bounds in the stability problem of swirling flows

Upper and lower bounds on the wave velocity of axisymmetric neutral modes in the stability problem of inviscid swirling flows are obtained.

Assessment of artificial fluid properties for high-order accurate large-eddy simulations of shock-free compressible turbulent flows with strong temperature gradients

The capability of artificial fluid properties (AFP) to serve as a sub-grid scale (SGS) model and to provide stability in high-order accurate large-eddy simulations (LES) of shock-free compressible turbulent flows with strong temperature gradients is assessed. Predictive LES of such flows, which are directly relevant to applications such as engines, require an accurate description of turbulence, which typically makes them susceptible to issues of accuracy and numerical instabilities. In the present implementation, AFP schemes along with a high-order compact filter ensure robustness of otherwise non-dissipative high-order central difference spatial discretisation schemes. It is shown that a judicious choice of both filter and AFP parameters is required in order for AFP to be used as an accurate SGS model. Optimal parameter values for the AFP and the corresponding filter parameter are identified and validated in the framework of the current LES solver. First, an inviscid Taylor-Green vortex configuration is examined to evaluate the ideal combination of the filter parameter and the coefficients in the AFP framework. It is verified that AFP works as an alternative model for the unclosed subgrid-scale (SGS) dissipation terms. The AFP results are also compared to those from the dynamic Smagorinsky model (DSM), with good agreement. Second, the optimal parameter values of AFP obtained from the inviscid Taylor-Green vortex tests are validated on a series of test configurations with increasing complexity, that include an inviscid 1D advection problem involving a thermal contact discontinuity, an inviscid 2D Kelvin-Helmholtz instability test and 3D experimental unheated and heated spatially developing turbulent round jets. Robust and accurate predictions are shown with the LES solver employing the optimal model parameters in each of these tests. In particular, good agreement between the LES predictions and experimental measurements for both the flow and temperature statistics is achieved for the laboratory jets, demonstrating the accuracy of the LES/AFP simulations applied to practical turbulent configurations. The robustness of the solver in the presence of strong temperature gradients is also demonstrated by simulating a round jet case having very high temperature gradients similar to those occurring in reacting flows. In contrast, the LES/DSM simulation of this case is noted to be unstable, and failed to provide converged results.

A higher-order error estimation framework for finite-volume CFD

Computational fluid dynamics is an invaluable tool for both the design and analysis of aerospace vehicles. Reliable error estimation techniques are needed to ensure that simulation results are accurate enough to be used in engineering decision-making processes. In this work, a framework for estimating error and improving solution accuracy is presented. A linearized error transport equation (ETE) is used to estimate local discretization errors. A truncation error estimation technique is proposed which combines aspects of higher-order residual methods and continuous residual methods. The equivalence between adjoint and ETE methods for functional error estimation is demonstrated. Using adjoint/ETE equivalence, the higher-order properties of adjoint methods are extended to ETE methods. Consequently, ETE error estimates are shown to converge to the true discretization error at a higher-order rate. ETE error estimates are then used to correct the entire primal solution, and by extension, all output functionals, to higher order. The computational advantages of this ETE approach are discussed. Results are presented for 1D and 2D inviscid and viscous flow problems on grids with smoothly varying and non-smoothly varying grid metrics.

A low-dissipation finite element scheme for scale resolving simulations of turbulent flows

The present work extends the conservative convective scheme proposed by Charnyi et al. (2017) [13], originally formulated for mixed finite elements and tested in laminar flows, to equal order finite elements. A non-incremental fractional-step method is used to stabilise pressure, allowing the use of finite element pairs that do not satisfy the inf-sup conditions, such as equal order interpolation for the velocity and pressure used in this work. The final scheme preserves momentum and angular momentum at the discrete level; the error in the conservation of kinetic energy introduced by this stabilisation is of O(δt,h2) in the case of linear finite elements. The low dissipation strategy is tested on a set of relevant turbulent cases. First, by using direct numerical simulation on the inviscid and viscous Taylor-Green vortex problem at Re=1600 and later, coupled with the Vreman (2004) [25] sub-grid stress model for performing large-eddy simulations on a turbulent channel flow at Reτ=950, the flow past a sphere at ReD=104 and the flow around an Ahmed body at ReH=2×105. In all cases the performance of the presented formulation is fairly good and it has been capable of reproducing the reference results with good accuracy.

On degenerate circular and shear flows: the point vortex and power law circular flows

We consider the problem of asymptotic stability and linear inviscid damping for perturbations of a point vortex and similar degenerate circular flows. Here, key challenges include the lack of strict monotonicity and the necessity of working in weighted Sobolev spaces whose weights degenerate as the radius tends to zero or infinity. By using a Fourier multiplier approach, we construct energy functionals to deduce stability in a perturbative setting. For sufficiently high spherical harmonics, we can handle any circular flows with power law singularities or zeros as or , while for low frequencies we can treat circular flows close to the Taylor–Couette flow. Similar results apply in the planar shear flow case close to Couette.

Modeling of interaction of long-wave disturbances with a shock wave on a flat plate with allowance for real gas effects

The problem of the long-wave disturbance interaction with a shock wave on a wedge at an angle of attack is considered. Numerical simulations are performed by solving two-dimensional unsteady Navier–Stokes equations with the module addition created by the user and allowing taking into account the vibrational nonequilibrium of carbon dioxide molecules within the two-temperature model of relaxation flows. A parametric study on the effect of Mach numbers, Reynolds numbers and angles of attack on the disturbance transformation coefficients is carried out for the following gases: equilibrium air, equilibrium vibrationally excited CO2, and nonequilibrium CO2. The calculation results on the pressure pulsations on the model surface were compared with the data obtained for long-wave disturbance from the general solution of the inviscid problem of the disturbance interaction with a shock wave on a wedge [1]. It is shown that the obtained long wavelength disturbance transformation coefficients are practically the same as for the inviscid case in the studied range of Mach numbers (M∞=4.5÷8.9).

An enthalpy‐preserving shock‐capturing term for residual distribution schemes

SummaryIn this contribution, we investigate strategies to perform shock‐capturing computation of steady hypersonic flow fields by means of residual distribution schemes. The ultimate objective is the computation of flow solutions for which the correct upstream enthalpy value is recovered in the postshock region. To this end, the parallelism existing between the classical Bx scheme and the stabilized finite element techniques is exploited. The simple Lax‐Friedrichs dissipation term is leveraged to build two new residual distribution schemes. Upon testing on both inviscid and viscous steady problems, solutions obtained with one of the two schemes are shown to recover the correct upstream total enthalpy level in the postshock region. This last scheme provides also improved wall pressure and skin friction predictions; heat transfer predictions are, unfortunately, similar to those offered by the Bx scheme. A conjecture for explaining this behavior is exposed.

A comprehensive high-order solver verification methodology for free fluid flows

The aim of this article is to present a comprehensive methodology for the verification of computational fluid dynamics (CFD) solvers with a special attention to aspects pertinent to discretizations with orders of accuracy (OOAs) higher than two. The method of manufactured solutions (MMS) is adopted and a series of manufactured solutions (MSs) is introduced that examines various components of CFD solvers for free flows (not bounded by walls), including inviscid, laminar and turbulent problems when the latter are modeled by the Reynolds-averaged Navier–Stokes (RANS) equations. The treatment of curved elements is also examined. These MSs are furthermore conceived with demonstrated suitability for the verification of OOAs up to the sixth. Each MS is as well utilized to discuss salient aspects useful to the code verification methodology such as the relative qualities of the most useful norms in measuring the discretization error, the sensitivity analysis of the verification process to forcing function terms, the relation between residual minimization and discretization error convergence in iterative solutions and finally the sensitivity of high-order discretizations to grid stretching and self-similarity. Furthermore, scripts and code are provided as accompanying material to assist the interested reader in reproducing the verification results of each manufactured solution (MS).

A robust and efficient finite volume method for compressible inviscid and viscous two-phase flows

A robust and efficient density-based finite volume method is developed for solving the six-equation single pressure system of two-phase flows at all speeds on hybrid unstructured grids. Unlike conventional approaches where an expensive exact Riemann solver is normally required for computing numerical fluxes at the two-phase interfaces in addition to AUSM-type fluxes for single-phase interfaces in order to maintain stability and robustness in cases involving interactions of strong pressure and void-fraction discontinuities, a volume-fraction coupling term for the AUSM+-up fluxes is introduced in this work to impart the required robustness without the need of the exact Riemann solver. The resulting method is significantly less expensive in regions where otherwise the Riemann solver would be invoked. A transformation from conservative variables to primitive variables is presented and the primitive variables are then solved in the implicit method in order for the current finite volume method to be able to solve, effectively and efficiently, low Mach number flows in traditional multiphase applications, which otherwise is a great challenge for the standard density-based algorithms. A number of benchmark test cases are presented to assess the performance and robustness of the developed finite volume method for both inviscid and viscous two-phase flow problems. The numerical results indicate that the current density-based method provides an attractive and viable alternative to its pressure-based counterpart for compressible two-phase flows at all speeds.

Low mass planet migration in Hall-affected disks

Recent developments in non-ideal magnetohydrodynamic simulations of protoplanetary disks suggest that instead of being traditional turbulent (viscous) accretion disks, they have a largely laminar flow with accretion driven by large-scale wind torques. These disks are possibly threaded by Hall-effect generated large-scale horizontal magnetic fields. We have examined the dynamics of the corotation region of a low mass planet embedded in such a disk and the evolution of the associated migration torque. These disks lack strong turbulence and associated turbulent diffusion, and the presence of a magnetic field and radial gas flow presents a situation outside the applicability of previous corotation torque theory. We summarize the analytical analysis of the corotation torque, give details on the numerical methods used, and in particular the relative merits of different numerical schemes for the inviscid problem.

Fluid sloshing in rectangular vessels with side-wall baffles using conformal mappings of multiply-connected domains

This paper focuses on the problem of inviscid, irrotational, incompressible fluid sloshing in a rectangular vessel with rigid, impermeable side-wall baffles, and investigates the feasibility of using time-dependent conformal mappings to numerically simulate the evolution of the unknown free-surface in fully-dynamic simulations. An algorithm which uses conformal mappings of a multiply-connected domain to relate the conjugate harmonic functions along the free-surface is documented, and kinematic results presented for a prescribed free-surface motion. The results show that the specific mapping for an infinite depth fluid has one free, within specific bounds, mapping parameter, while the mapping for finite depth fluids has two free mapping parameters. It is shown that having two free parameters gives a wider range of situations under which the conformal mapping can be computed, and it is concluded that the finite depth mapping should be used (in the appropriate limit) even for infinite depth simulations. Overall it is found that a computationally efficient algorithm can be devised to relate the conjugate harmonic functions along the free-surface of the flow domain.

A third-order gas-kinetic CPR method for the Euler and Navier–Stokes equations on triangular meshes

A third-order accurate gas-kinetic scheme based on the correction procedure via reconstruction (CPR) framework is developed for the Euler and Navier–Stokes equations on triangular meshes. The scheme combines the accuracy and efficiency of the CPR formulation with the multidimensional characteristics and robustness of the gas-kinetic flux solver. Comparing with high-order finite volume gas-kinetic methods, the current scheme is more compact and efficient by avoiding wide stencils on unstructured meshes. Unlike the traditional CPR method where the inviscid and viscous terms are treated differently, the inviscid and viscous fluxes in the current scheme are coupled and computed uniformly through the kinetic evolution model. In addition, the present scheme adopts a fully coupled spatial and temporal gas distribution function for the flux evaluation, achieving high-order accuracy in both space and time within a single step. Numerical tests with a wide range of flow problems, from nearly incompressible to supersonic flows with strong shocks, for both inviscid and viscous problems, demonstrate the high accuracy and efficiency of the present scheme.