Toward robust linear implicit schemes for steady state hypersonic flows

This paper develops a numerical procedure to accelerate the convergence of the Favre-averaged non-linear harmonic (FNLH) method. The scheme provides a unified mathematical framework for solving the sparse linear systems formed by the mean flow and the time-linearized harmonic flows of FNLH in an explicit or implicit fashion. The approach explores the similarity of the sparse linear systems of FNLH and leads to a memory-efficient procedure, so that its memory consumption does not depend on the number of harmonics to compute. The proposed method has been implemented in the industrial computational fluid dynamics solver Hydra. Three test cases are used to conduct a comparative study of explicit and implicit schemes in terms of convergence, computational efficiency, and memory consumption. Comparisons show that the implicit scheme yields better convergence than the explicit scheme and is also roughly 7–10 times more computationally efficient than the explicit scheme with four levels of multigrid. Furthermore, the implicit scheme consumes only approximately 50% of the memory required by the explicit scheme with four levels of multigrid. Compared with the full-annulus unsteady Reynolds-averaged Navier–Stokes simulations, the implicit scheme produces comparable results to URANS with computational time and memory consumption that are two orders of magnitude smaller.

Over the past decade, finite volume schemes have significantly advanced, becoming well-regarded for solving linear and nonlinear population balance equations (PBEs). These schemes are highly accurate and efficient, making them ideal for applications like liquid–liquid dispersion, bubble and droplet fragmentation, in the chemical and pharmaceutical industries. Solving PBEs in continuous form remains challenging, particularly with complex fragmentation kernels and selection functions. Typically, these problems are tackled by forming discretized ordinary differential equations, with accuracy depending on the mesh type and cell count. To address these challenges, a new semi-analytical approach for solving the weighted finite volume scheme breakage equations has been developed [Kumar et al. (2015), “Development and convergence analysis of a finite volume scheme for solving breakage equation,” SIAM J. Numer. Anal. 53(4), 1672–1689]. This approach can replace traditional numerical schemes using the fourth order Runge-Kutta method. The mesh-independence with respect to time of finite volume schemes allows efficient coupling with computational fluid dynamics (CFD) tools. The accuracy and efficiency of the proposed method have been validated with analytically tractable and physically relevant fragmentation kernels and selection functions, demonstrating high accuracy in estimating number density functions and their integral moments. This new approach reduces computational time by approximately 60%, making it an excellent option for integration with CFD software due to its efficiency.

We propose a cell-centered finite volume (FV) scheme with the minimal stencil formed by the closest neighbouring cells. The discrete solution satisfies the discrete maximum principle and approximates the exact solution with second-order accuracy. The coefficients in the FV stencil depend on the solution; therefore, the FV scheme is nonlinear. The scheme is applied to a steady state advection-diffusion equation discretized on a general polygonal mesh.

A numerical solution for a coupled hyperbolic thermoelasticity problem of the Lord‐Shulman type is presented. An infinite layer with free boundaries that are kept at a constant temperature is irradiated by a short laser impulse. In this case the problem can be studied in the one‐dimensional formulation. An impact of the laser impulse on a medium is modeled by an internal heat source. The laser impulse intensity decays with a distance from an irradiated surface according to the Bouguer law. The layer is in an unperturbed state at the initial moment of time. The material properties of copper are taken to be the thermo‐mechanical properties of the layer.Two approaches for the boundary value problem are considered: the finite‐difference method (FDM) and the finite‐volume method (FVM). In the first case the system of two second‐order PDE is investigated, in the second case a system of integral balance equations and an integral thermal conductivity equation are composed. To integrate these equations both the explicit and implicit schemes are used. The obtained results are compared with each other and with an analytical solution. A numerical error is estimated for different time and space step sizes. The features inherent in various ways of integration of the coupled hyperbolic thermoelastic problem with the thermal excitation are found.

The heat conduction equation is a combination of first-order and second-order differential equations. Solving first-order differential equations is necessary to examine temperature as a function of time. Meanwhile, solving second-order differential equations is needed to examine temperature as a function of space. The heat flux equation is based on Fourier's law, which shows that temperature is a function of time and space. Understanding heat conduction can be improved by building a heat propagation model on the Solar ERK dryer plate. Analysis of heat propagation on the drying plate used the Finite Difference Approach (FDA) method with explicit and implicit schemes. With an explicit scheme, the FDA method calculates the temperature (T) at a point on the spatial derivative term, when T is at time t, while the implicit scheme calculates T at a point on the space derivative term when T is at time t+Δt. Heat propagation at each time change was analyzed by developing a program using the MATLAB 17 application. The results of the analysis show that there are differences in heat propagation between the explicit and implicit schemes. The convergence and stability of calculations in explicit schemes are unstable, causing problems at the time step. Meanwhile, the implicit scheme is carried out simultaneously on all nodes so that convergence and stability are easily maintained, and there are no time-step limitations.

Second-order accurate finite volume schemes with the discrete maximum principle for solving Richards’ equation on unstructured meshes

This paper presents an investigation of Reynolds-averaged Navier-Stokes (RANS) turbulence models used in computational fluid dynamics (CFD) simulations of boundary layer flow and heat transfer in high Mach number flows. This study evaluates an industry standard RANS turbulence model (k-ω SST [1]) and a recently proposed modification to that model [2], for prediction of the mean velocity field in selected test cases. The test cases were previously documented by Duan et al. [3], who used direct numerical simulation (DNS) to calculate boundary layer flow of an ideal gas over a flat plate at freestream Mach numbers ranging from 2.5 to 14 and wall to recovery temperature ratios of 0.18 to 1. Boundary layer profiles were evaluated at two streamwise locations, one where the boundary layer height matched the DNS data and the second where the wall shear stress matched DNS data. Results show that the accuracy of RANS models degrades for high-speed regimes compared to incompressible or subsonic flow but that the compressibility correction factor [2] improves the results for some of the test cases.

Many plasma physics models have been proved to possess a non-canonical hamiltonian structure. Invariants like the hamiltonian or Casimir invariants like div B = 0 follow immediately from this structure. Hence discretizing the infinite dimensional hamiltonian structure so that we obtain a finite dimensional hamiltonian structure provides a natural way to conserve discrete invariants. After a short review of geometric discretization for kinetic and hybrid fluid-kinetic models, we will focus on structure preserving discretization of compressible ideal MHD. In order to robustly handle shocks and also keep the symmetries inherent to the MHD model, we will introduce a semi-implicit hybrid model coupling Finite Element Exterior Calculus and Finite Volume schemes. A splitting approach is designed so that we may take advantage of the conservation properties and robustness of the Finite Volume schemes for the non-linear advection, while relying on a structure-preserving discretization of the magneto-acoustic terms based on Finite Element Exterior calculus. Moreover, the nonlinear convective terms are treated via an explicit time-discretization, while the magnetic and acoustic terms are solved via an implicit time-discretization. Thanks to this, the resulting CFL condition will depend, at least formally, only on the fluid velocity and not on the Alfvén or sound speeds that may become too stringent in the low Mach regimes. In this approach, the divergence free constraint of the magnetic field is always preserved up to machine precision, and the symmetry of the physical model is also reflected to the symmetry of the final algebraic nonlinear systems that are solved in an implicit step. Thanks to the symmetry of the systems, the very efficient matrix-free conjugate gradient method may be employed.

PurposeTo present conversion of the advection upwind splitting method (AUSM+) from the conventional density‐based and coupled formulation to the pressure‐based and segregated formulation.Design/methodology/approachThe spatial discretization is done by a finite volume method. A collocated grid cell‐center formulation is used. The pressure‐correction procedure is set up in the usual way for a compressible flow problem. The conventional Rhie‐Chow interpolation methodology for the determination of the transporting velocity, and the conventional central interpolation for the pressure at the control volume faces, are replaced by AUSM+ definitions.FindingsThe AUSM+ flux definitions are spontaneously well suited for use in a collocated pressure‐correction formulation. The formulation does not require extensions to these flux definitions. As a consequence, the results of a density‐based fully coupled method, are identical to the results of a pressure‐based segregated formulation. The advantage of the pressure‐correction method with respect to the density‐based method, is the higher efficiency for low Mach number applications. The advantage of the AUSM+ flux definition for the transporting velocity with respect to the conventional Rhie‐Chow interpolation, is the improved accuracy in high Mach number flows. As a consequence, the combination of AUSM+ with a pressure‐correction method leads to an algorithm with improved performance for flows at all Mach numbers.Originality/valueA new methodology, with obvious advantages, is composed by the combination of ingredients from an existing spatial discretization method (AUSM+) and an existing time stepping method (pressure‐correction).

The Finite Volume Method (FVM) was introduced into the field of computational fluid dynamics in the beginning of the seventies (McDonald 1971, Mac-Cormack and Paullay 1972). From the physical point of view the FVM is based on balancing fluxes through control volumes, i. e. the Eulerian concept is used (see section 1.1.4). The integral formulation of conservative laws are discretized directly in space. From the numerical point of view the FVM is a generalization of the FDM in a geometric and topological sense, i. e. simple finite volume schemes can be reduced to finite difference schemes. The FDM is based on nodal relations for differential equations, whereas the FVM is a discretization of the governing equations in integral form. The Finite Volume Method can be considered as specific subdomain method as well. FVM has two major advantages: First, it enforces conservation of quantities at discretized level, i. e. mass, momentum, energy remain conserved also at a local scale. Fluxes between adjacent control volumes are directly balanced. Second, finite volume schemes takes full advantage of arbitrary meshes to approximate complex geometries. Experience shows that non-conservative schemes are generally less accurate than conservative ones, particularly in the presence of strong gradients.

A novel computational approach to combine the optical and thermal modelling of Linear Fresnel Collectors using the finite volume method

The new generation of power plants based on innovative thermodynamic cycles operating with unconventional working fluids, such as CO2 close to its thermodynamic critical point or organic fluids close to their vapour saturation line, is an attractive option for high efficiency conversion of sustainable energy sources in the low and medium power output range. One of the most challenging tasks is to make this technology competitive by realizing highly efficient turbomachinery components. The main difficulty stems from the strongly non-ideal fluid dynamic behaviour of such fluids, which largely deviates from that of fluids commonly employed in standard applications. As a consequence, special design is needed, as available methods for turbomachinery of steam or gas turbine power plants cannot be used. The scarce knowledge on both the numerical modeling of non-ideal fluid flows and on their performance in turbomachinery applications makes research crucial. Computational fluid dynamics (CFD) is a powerful tool to investigate the physics of non-ideal flows and to assist the design of unconventional machines. This thesis presents original research in the field of non-ideal fluid dynamics on three complementary aspects, namely numerical methods, applications, and theory, aiming at (i) developing new numerical schemes for the accurate and efficient CFD simulation of non-ideal compressible flows; (ii) investigating the performance of turbomachinery operating with unconventional fluids; (iii) studying the hydrodynamic stability of variable property fluid flows. The core of the report consists of five self-contained chapters, each addressing a specific problem, which are grouped into three parts according to their area of contribution. In the first part, novel implicit schemes for the spatial discretization of the Navier--Stokes equations in the framework of the finite-volume method are introduced (Chapter 2). The look-up table approach is extensively analyzed and proposed as an accurate and efficient alternative to the direct solution of the equation of state model for fluid property evaluation. In order to address the proper handling of sub-domains interfaces needed in unsteady turbomachinery simulations, a new flux-conserving treatment of non-conformal mesh blocks interfaces is proposed in Chapter 3. In the second part, the focus shifts to the investigation of non-ideal fluid flows in turbomachines. Chapter 4 reports the CFD computation of the performance map of a 20 mm diameter radial compressor operating with supercritical CO2 designed for a power of 50 kW and a rotational speed of 75 krpm. Results obtained by means of Reynolds-averaged Navier--Stokes (RANS) simulations are compared to experimental measurements of isentropic efficiency and specific enthalpy rise, showing a reasonable agreement and providing a validation for the numerical schemes introduced in Chapter 2. Chapter 5 investigates the unsteady operation of a supersonic turbine for organic Rankine cycle applications using unsteady RANS simulations. The evolution of shock waves and viscous wakes and their interaction are described in details and related to the fluctuations of blade loads, which can be of the same order of magnitude of their time-averaged values, demonstrating the primary importance of performing unsteady simulations when studying such non-conventional machines. A third part concludes the thesis presenting an analytical study of the hydrodynamic stability of a thermally-stratified channel flow with temperature dependent properties. The classical linear stability analysis framework is extended to include the effect of variable transport properties. Various distributions of thermal conductivity and specific heat across the channel height are considered in order to assess under which conditions variable properties contribute to the stabilization or destabilization of the flow. The presented results have practical implications in the study of the transition to turbulence of wall bounded flows of unconventional fluids and in devising flow control strategies.

Non-equilibrium effects play a vital role in high-speed and rarefied gas flows and the accurate simulation of these flow regimes are far beyond the capability of near-local-equilibrium Navier-Stokes-Fourier equations. Eu proposed generalized hydrodynamic equations which are consistent with the laws of irreversible thermodynamics to solve this problem. Based on Eu’s generalized hydrodynamics equations, a computation model, namely the nonlinear coupled constitutive relations (NCCR), was developed by R.S. Myong and applied successfully to one-dimensional shock wave structure and two-dimensional rarefied flows. In this paper, finite volume schemes, including LU-SGS time advance scheme, MUSCL interpolation and AUSMPW+ scheme, are firstly adopted to investigate NCCR model’s validity and potential in three-dimensional complex hypersonic rarefied gas flows. Moreover, in order to solve the computational stability problems in 3D complex flows, a modified solution is developed for the NCCR model. Finally, the modified solution is tested for a slip complex flow over a 3D hollow cylinder-flare configuration. The numerical results show that the NCCR model by the modified solution yields good solutions in better agreement with the DSMC results and experimental data than NSF equations, and imply NCCR model’s great potential capability in further application.

While the SIMPLE algorithm is most widely used for the simulations of flow phenomena that take place in the industrial equipments or the manufacturing processes, it is less adopted for the simulations of the free surface flow. Though the SIMPLE algorithm is free from the limitation of time step, the free surface behavior imposes the restriction on the time step. As a result, the explicit schemes are faster than the implicit scheme in terms of computation time when the same time step is applied to, since the implicit scheme includes the numerical method to solve the simultaneous equations in its procedure. If the computation time of SIMPLE algorithm can be reduced when it is applied to the unsteady free surface flow problems, the calculation can be carried out in the more stable way and, in the design process, the process variables can be controlled based on the more accurate data base. In this study, a modified SIMPLE algorithm is presented fur the free surface flow. The broken water column problem is adopted for the validation of the modified algorithm (MoSIMPLE) and for comparison to the conventional SIMPLE algorithm.

This paper presents a partitioned fluid–structure interaction (FSI) solver to model elastohydrodynamic lubrication (EHL) of line contacts. The FSI model was constructed using the multiphysics simulation software ansys, wherein an iterative implicit coupling scheme is implemented to facilitate the interaction between fluid and solid components. The model uses a finite volume method (FVM) based computational fluid dynamics (CFD) solver to determine the lubricant flow behavior using the Navier–Stokes equations. Additionally, the finite element method (FEM) is utilized to model the structural response of the solid. Fluid cavitation, compressibility, non-Newtonian lubricant rheology, load balance algorithm, and dynamic meshing were incorporated in the FSI model. The pressure and film thickness results obtained from the model are presented for a wide range of loads, speeds, slide to roll ratios (SRR), surface dent, material properties (elastic plastic), etc. The model presents a detailed understanding of EHL contacts by removing any assumptions relative to the Reynolds equation. It provides the (i) two-dimensional variation of pressure, viscosity, etc., in the fluid and (ii) stress, elastic/plastic strain in the solid, simultaneously. The FSI model is robust, easy to implement, and computationally efficient. It provides an effective approach to solve sophisticated EHL problems. The FSI model was used to investigate the effects of surface dents, plasticity and material inclusions under heavily loaded lubricated line contacts as can be found in gears and rolling element bearings. The results from the model exhibit excellent corroboration with published results based on the Reynolds equation solvers.