Multidimensional Flows Research Articles

Quantum algorithm for solving the advection equation using Hamiltonian simulation

A quantum algorithm for solving the advection equation by embedding the discrete time-marching operator into Hamiltonian simulations is presented. One-dimensional advection can be simulated directly since the central finite-difference operator for first-order derivatives is anti-Hermitian. Here this is extended to industrially relevant multidimensional flows with realistic boundary conditions and arbitrary finite-difference stencils. A single copy of the initial quantum state is required and the circuit depth grows linearly with the required number of time steps, the sparsity of the time-marching operator, and the inverse of the allowable error. State-vector simulations of a scalar transported in a two-dimensional channel flow and lid-driven cavity configuration are presented as a proof of concept of the proposed approach. Published by the American Physical Society 2024

A viscoelastic flow model of Maxwell-type with a symmetric-hyperbolic formulation

Maxwell models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But the usual Maxwell models allow one to define well motions mostly for one-dimensional flows only. To define unequivocal multi-dimensional viscoelastic flows (as solutions to well-posed initial-value problems) we advocated in [ESAIM:M2AN 55 (2021), p. 807-831] an upper-convected Maxwell model for compressible flows with a symmetric-hyperbolic formulation. Here, that model is derived again, with new details.

An efficient forward semi-Lagrangian model

An efficient forward trajectory model is proposed, in which the property and position of the fluids advected from the Euler coordinates to the Lagrangian coordinates can be accurately evaluated. After sorting and aligning those fluid elements on the irregular Lagrangian curves, we apply the cubic or other high-degree polynomials to interpolate the properties of the elements from the irregular curves to the regular grids. There is no need to solve the cubic equations and the associated coefficients as proposed previously. The model is quite simple, accurate, and much more efficient than the previous models. It also allows higher-order polynomials to be employed in the interpolations. It is suitable for simulating the multi-dimensional fast-moving flows with large Courant Numbers, the transport of pollutants in the atmosphere and ocean, and movement of raindrops in atmospheric models.

A class of symmetric-hyperbolic PDEs modelling fluid and solid continua

We generalize a new symmetric-hyperbolic system of PDEs proposed in [ESAIM:M2AN 55 (2021) 807-831] for Maxwell fluids to a class of systems that define unequivocally multi-dimensional visco-elastic flows. Precisely, within a general setting for continuum mechanics, we specify constitutive assumptions i) that ensure the unequivocal definition of motions satisfying widely-admitted physical principles, and ii) that contain [ESAIM:M2AN 55 (2021) 807-831] as one particular realization of those assumptions. The new class can capture the mechanics of various materials, from solids to viscous fluids, possibly with temperature dependence and heat conduction.

On the convergence rates of multi-dimensional subsonic irrotational flows in unbounded domains

On the convergence rates of multi-dimensional subsonic irrotational flows in unbounded domains

Global strong solutions for the multi-dimensional compressible viscoelastic flows with general pressure law

In this paper, we mainly focus on the compressible viscoelastic flows of Oldroyd type with the general pressure law, with one of the non-Newtonian fluids exhibiting the elastic behavior. For the viscoelastic flows of Oldroyd type with the general pressure law, P′(ρ̄)+α>0, with α > 0 being the elasticity coefficient of the fluid, we prove the global existence and uniqueness of the strong solution in the critical Besov spaces when the initial data u⃗0 and the low frequency part of ρ0, τ0 are small enough compared to the viscosity coefficients. In particular, when the viscosity is large, the part of the initial data can be large. The proof we display here does not need any compatible conditions. In addition, we also obtain the optimal decay rates of the solution in the Besov spaces.

Efficient Finite Difference WENO Scheme for Hyperbolic Systems with Non-conservative Products

Higher order finite difference Weighted Essentially Non-Oscillatory (WENO) schemes have been constructed for conservation laws. For multidimensional problems, they offer high order accuracy at a fraction of the cost of a finite volume WENO or DG scheme of comparable accuracy. This makes them quite attractive for several science and engineering applications. But, to the best of our knowledge, such schemes have not been extended to non-linear hyperbolic systems with non-conservative products. In this paper, we perform such an extension which improves the domain of applicability of such schemes. The extension is carried out by writing the scheme in fluctuation form. We use the HLLI Riemann solver of Dumbser and Balsara (2016) as a building block for carrying out this extension. Because of the use of an HLL building block, the resulting scheme has a proper supersonic limit. The use of anti-diffusive fluxes ensures that stationary discontinuities can be preserved by the scheme, thus expanding its domain of applicability. Our new finite difference WENO formulation uses the same WENO reconstruction that was used in classical versions, making it very easy for users to transition over to the present formulation. For conservation laws, the new finite difference WENO is shown to perform as well as the classical version of finite difference WENO, with two major advantages:- 1) It can capture jumps in stationary linearly degenerate wave families exactly. 2) It only requires the reconstruction to be applied once. Several examples from hyperbolic PDE systems with non-conservative products are shown which indicate that the scheme works and achieves its design order of accuracy for smooth multidimensional flows. Stringent Riemann ... *Abstract truncated, see PDF*

Data-Driven High-to-Low for Coarse Grid System Thermal Hydraulics

Traditional one-dimensional system thermal-hydraulic analysis has been widely applied in the nuclear industry for licensing purposes because of its numerical efficiency. However, such tools have inherently limited opportunities for modeling multiscale multidimensional flows in large reactor enclosures. Recent interest in three-dimensional coarse grid (CG) simulations has shown their potential in improving the predictive capability of system-level analysis. At the same time, CGs do not allow one to accurately resolve and capture turbulent mixing and stratification, whereas implemented in CG solvers relatively simple turbulence models exhibit large model form uncertainties. Therefore, there is a strong interest in further advances in CG modeling techniques. In this work, two high-to-low data-driven (DD) methodologies (and their combination) are explored to reduce grid and model-induced errors using a case study based on the Texas A&M upper plenum of a high-temperature gas-cooled reactor facility. The first approach relies on the use of a DD turbulence closure [eddy viscosity predicted by a neural network (NN)]. A novel training framework is suggested to consider the influence of grid cell size on closure. The second methodology uses a NN to predict velocity errors to improve low-fidelity results. Both methodologies and their combination have shown the potential to improve CG simulation results by using data with higher fidelity.

Context-Aware Clustering and the Optimized Whale Optimization Algorithm: An Effective Predictive Model for the Smart Grid

For customers to participate in key peak pricing, period-of-use fees, and individualized responsiveness to demand programmes taken from multi-dimensional data flows, energy use projection and analysis must be done well. However, it is a difficult study topic to ascertain the knowledge of use of electricity as recorded in the electricity records' Multi-Dimensional Data Streams (MDDS). Context-Aware Clustering (CAC) and the Optimized Whale Optimization Algorithm were suggested by researchers as a fresh power usage knowledge finding model from the multi-dimensional data streams (MDDS) to resolve issue (OWOA). The proposed CAC-OWOA framework first performs the data cleaning to handle the noisy and null elements. The predictive features are extracted from the novel context-aware group formation algorithm using the statistical context parameters from the pre-processed MDDS electricity logs. To perform the energy consumption prediction, researchers have proposed the novel Artificial Neural Network (ANN) predictive algorithm using the bio-inspired optimization algorithm called OWOA. The OWOA is the modified algorithm of the existing WOA to overcome the problems of slow convergence speed and easily falling into the local optimal solutions. The ANN training method is used in conjunction with the suggested bio-inspired OWOA algorithm to lower error rates and boost overall prediction accuracy. The efficiency of the CAC-OWOA framework is evaluated using the publicly available smart grid electricity consumption logs. The experimental results demonstrate the effectiveness of the CAC-OWOA framework in terms of forecasting accuracy, precision, recall, and duration when compared to underlying approaches.

Space-Time Resource Integrated Optimization Method for Time-of-Day Division at Intersection Based on Multidimensional Traffic Flows

Based on the change trends of traffic flow in different controlled directions at an intersection, the space-time resource integrated optimization method for TOD (time-of-day) division based on multidimensional traffic-flow data is proposed in this paper. By analyzing the traffic-flow data of 8, 4, 2, and 1 dimensions commonly used at the intersection, the dynamic Fisher algorithm is used to complete the time segment division of the traffic-flow sequence of different dimensions. On this basis, the preliminary TOD division is completed, and the phase timing and lane-use assignment corresponding to the preliminary time periods are optimized. Then, the adjacent time periods are merged and tested to complete the final result of the TOD division. In order to verify the effectiveness of the proposed method, the schemes based on traffic-flow data of different dimensions are carried out by using the data at an actual intersection in Wuhu City, and the total and average vehicle delays of different schemes throughout the day are evaluated by VISSIM. The results show that the more dimensions of traffic flow data are adopted, the more refined the TOD division scheme is, and the less the total delay and average delay at intersections throughout the day are. In particular, the TOD division scheme after further optimizing the lane-use assignment can further reduce the total and average vehicle delay throughout the day. It shows that the method using multidimensional traffic-flow data at an intersection to carry out the integrated optimization of TOD division, lane-use assignment, and phase timing has good applicability.

Development of a carbuncle-free and low-dissipation Roe-type scheme: Applications to multidimensional Euler flows

The Roe scheme is known to have high resolution but to be afflicted by the notorious shock anomalies (such as the carbuncle phenomena) and the unphysical expansion shocks. This paper aims to develop a new Roe-type scheme which is robust for strong shock waves and low dissipative for contact waves and shear layers. Based on the mechanism analysis of shock instability, a dissipation-controlling strategy is adopted to adjust the entropy wave viscosity and shear wave viscosity of the transverse numerical flux in the numerical shock layer and the robustness of the Roe scheme is thus enhanced. The violation of entropy condition and the problem of expansion shocks are also fixed to broaden the scheme’s application scope. In addition, the THINC reconstruction procedure and a BVD algorithm are implemented to further improve the resolution for contact waves and shear layers. Numerical results of a series of classical one-, two- and three-dimensional test problems fully demonstrate that the proposed new Roe-type scheme has not only better robustness for strong shock waves but also higher resolution for contact waves and shear layers than the original Roe scheme.

Global existence in critical spaces for non Newtonian compressible viscoelastic flows

We are interested in the multi-dimensional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the “div-curl” structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. In absence of compatible conditions, the new effective flux is introduced, which enables us to capture the dissipation arising from combination of density and deformation tensor. The partial dissipation in non-Newtonian compressible fluids, is weaker than that of classical Navier-Stokes equations.

On the validation of ATHLET 3-D features for the simulation of multidimensional flows in horizontal geometries under single-phase subcooled conditions

This paper provides an assessment of fluid transport and mixing processes inside the primary circuit of the test facility ROCOM through the numerical simulation of Test 2.1 with the system code ATHLET. The experiment represents an asymmetric injection of cold and non-borated water into the reactor coolant system (RCS) of a pressurized water reactor (PWR) to restore core cooling, an emergency procedure which may subsequently trigger a core re-criticality. The injection takes place at low velocity under single-phase subcooled conditions and presents a major challenge for the simulation in lumped parameter codes, due to multidimensional effects in horizontal piping and vessel arising from density gradients and gravity forces. Aiming at further validating ATHLET 3-D capabilities against horizontal geometries, the experiment conditions are applied to a ROCOM model, which includes a newly developed horizontal pipe object to enhance code prediction inside coolant loops. The obtained results show code strong simulation capabilities to represent multidimensional flows. Enhanced prediction is observed at the vessel inlet compared to traditional 1-D approach, whereas mixing overprediction from the descending denser plume is observed at the upper-half downcomer region, which leads to eventual deviations at the core inlet.

Secured energy ecosystems under Distributed Energy Resources penetration

Renewable energy systems have mushroomed in the form of microgrids and Virtual Power Plants (VPP). In Australia itself, there are over three million Distributed Energy Resources (DERs). Integrating these new energy ecosystems into current systems is becoming a horrendous task as multi-dimensional energy flows and new energy business models are evolving. The stakeholders in the energy sector are focused on reducing costs prematurely while integration standards are still evolving, and interoperability of Internet of things (IoT) with legacy systems has outstanding security concerns. In this paper, the changing energy landscape is examined, and cybersecurity issues associated with the operation of VPPs and renewable energy-based microgrids are highlighted. The security and operational scenarios of new ecosystems are outlined, with an emphasis on DER interoperability, security, and integration. The design and development of these evolving standards into these new energy ecosystems is detailed based on observations from experiments on real-world DER installations. This paper is an extension of work originally reported in proceedings of the 31st Australasian Universities Power Engineering Conference [1].

Application of the TV-HLL scheme to multidimensional ideal magnetohydrodynamic flows

Application of the TV-HLL scheme to multidimensional ideal magnetohydrodynamic flows

Core Invariants of the Mathematical Model of an Intelligent System

A holistic description of a universal mathematical model for the concept of an intelligent system is given. It is based on formalized invariants associated with the processes of creating and applying such systems. The core of the model is formed by consistent descriptions for sections of knowledge formalisms, components of multidimensional architecture and knowledge flows processes within it, as well as cybernetic hierarchical multi agents systems that control the intelligent systems subjective existence. The fundamental invariants of the knowledge presentation and processing are directly implemented by these main sections basic elements. Invariants form unified set of intelligent systems general attributes. This set allows carrying out comprehensive formal modelling of the intelligence. These invariants are associated with knowledge aspects. They are developed and used at knowledge areas that deal with exploring the memory structural organization and thinking processes models. The tools for transforming the proposed abstract model into the models of specific intelligent systems are morphisms of homomorphic expansion. These morphisms concretize the content of the main structural and functional elements of the intelligent system fundamental model. At the same time, the varieties of entities implemented by model elements are narrowed to the families of objects that make up applied intelligent systems. These systems inherit the properties of fundamental model common elements. Intermediate models of the processes of converting the original model into applied ones allow studying these models properties by mathematical tools. Intermediate models form the basis for the subsequent development of the technology of creating and applying multilevel intelligent systems.

An accurate and shock-stable genuinely multidimensional scheme for solving the Euler equations

In numerical simulations of multidimensional high Mach flows, the conventional low-diffusion upwind schemes built on the dimensional splitting method will encounter the shock instability which is mainly manifested as the infamous carbuncle phenomena. A linear stability analysis reveals that the shock instability is triggered by the unattenuated perturbations in the transverse direction of the flow field. In the present work, an accurate and shock-stable genuinely multidimensional numerical scheme based on the Toro-Vázquez splitting method is presented. Different from the conventional one-dimensional upwind schemes that just consider the waves propagating along the direction normal to an interface, the proposed multidimensional scheme whose multidimensional properties are achieved by calculating multidimensional numerical fluxes at each corner of an interface also takes into account the waves traveling along the transverse direction. For obtaining these multidimensional numerical fluxes at the corner, a simple multidimensional upwind method is used to solve the weakly hyperbolic convection subsystem and the pressure subsystem whose flux Jacobian has a complete set of linearly independent eigenvectors is calculated by a multidimensional HLLEM scheme. Based on Balsara’s framework for constructing multidimensional schemes, the total numerical flux across an interface is obtained by using the Simpson’s rule to assemble the conventional one-dimensional numerical flux with the multidimensional numerical fluxes at the corner. A series of benchmark test problems validate the accuracy, robustness and efficiency of the proposed multidimensional scheme.

A hybrid multidimensional Riemann solver to couple self-similar method with MULTV method for complex flows

Since proposed, the self-similarity variables based genuinely multidimensional Riemann solver is attracting more attentions due to its high resolution in multidimensional complex flows. However, it needs numerous logical operations in supersonic cases, which limit the method’s applicability in engineering problems greatly. In order to overcome this defect, a hybrid multidimensional Riemann solver, called HMTHS (Hybrid of MulTv and multidimensional HLL scheme based on Self-similar structures), is proposed. It simulates the strongly interacting zone by adopting the MHLLES (Multidimensional Harten-Lax-van Leer-Eifeldt scheme based on Self-similar structures) scheme at subsonic speeds, which is with a high resolution by considering the second moment in the similarity variables. Also, it adopts the MULTV (Multidimensional Toro and Vasquez) scheme, which is with a high resolution in capturing discontinuities, to simulate the flux at supersonic speeds. Systematic numerical experiments, including both one-dimensional cases and two-dimensional cases, are conducted. One-dimensional moving contact discontinuity case and sod shock tube case suggest that HMTHS can accurately capture one-dimensional expansion waves, shock waves, and linear contact discontinuities. Two-dimensional cases, such as the double Mach reflection case, the supersonic shock / boundary layer interaction case, the hypersonic flow over the cylinder case, and the hypersonic viscous flow over the double-ellipsoid case, indicate that the HMTHS scheme is with a high resolution in simulating multidimensional complex flows. Therefore, it is promising to be widely applied in both scholar and engineering areas.

The Half-Range Moment Method in Harmonically Oscillating Rarefied Gas Flows

The formulation of the half-range moment method (HRMM), well defined in steady rarefied gas flows, is extended to linear oscillatory rarefied gas flows, driven by oscillating boundaries. The oscillatory Stokes (also known as Stokes second problem) and the oscillatory Couette flows, as representative ones for harmonically oscillating half-space and finite-medium flow setups respectively, are solved. The moment equations are derived from the linearized time-dependent BGK kinetic equation, operating accordingly over the positive and negative halves of the molecular velocity space. Moreover, the boundary conditions of the “positive” and “negative” moment equations are accordingly constructed from the half-range moments of the boundary conditions of the outgoing distribution function, assuming purely diffuse reflection. The oscillatory Stokes flow is characterized by the oscillation parameter, while the oscillatory Couette flow by the oscillation and rarefaction parameters. HRMM results for the amplitude and phase of the velocity and shear stress in a wide range of the flow parameters are presented and compared with corresponding results, obtained by the discrete velocity method (DVM). In the oscillatory Stokes flow the so-called penetration depth is also computed. When the oscillation frequency is lower than the collision frequency excellent agreement is observed, while when it is about the same or larger some differences are present. Overall, it is demonstrated that the HRMM can be applied to linear oscillatory rarefied gas flows, providing accurate results in a very wide range of the involved flow parameters. Since the computational effort is negligible, it is worthwhile to consider the efficient implementation of the HRMM to stationary and transient multidimensional rarefied gas flows.

High‑Order Bicompact Schemes for Numerical Modeling of Multispecies Multi-Reaction Gas Flows

Euler equations for multidimensional inviscid gas flows with multispecies and multi‑reaction are considered. By using the Marchuk–Strang splitting method, an implicit numerical scheme for this system is constructed. Its convection part is computed using the bicompact scheme SDIRK3B4 of fourth order in space and third order in time, while its chemical part is computed using the L-stable Runge–Kutta method of second order. The SDIRK3B4 scheme is compared with the WENO5/SR scheme in the case of one- and two-dimensional flows with detonation waves. It is shown that the SDIRK3B4 scheme has the same actual accuracy as the WENO5/SR, but the former needs fewer time steps by a factor of 20–40 and does not require any special algorithms to suppress the nonphysical breakdown of detonation waves on relatively coarse meshes.