Fluid Boundary Conditions Research Articles

Regularity and uniqueness of solutions for compressible non-newtonian MHD equations with variable electrical conductivity and energy effects

This paper examines the existence, uniqueness, and regularity of solutions for a system of compressible non-Newtonian magnetohydrodynamics (MHD) equations with variable electrical conductivity and considering the energy equation. We establish the necessary conditions on fluid properties, external forces, and initial and boundary conditions to ensure the well-posedness of the system. Our results demonstrate the existence of regular solutions in appropriate functional spaces, the uniqueness of these solutions, and their higher regularity under smooth initial and boundary conditions. These findings provide a mathematical baseline for the study of complex interactions in compressible non-Newtonian MHD flows, and particularly the findings can be considered to support more accurate numerical simulations concerning a physical situation as well as deeper analytical solutions.

Heat transfer analysis in a horizontal anisotropic porous channel with Bi-viscous Bingham nanofluid and temperature-dependent Brownian diffusion

PurposeThe purpose of this article is to present a mathematical model for the fully developed flow of Bi-viscous Bingham nanofluid through a uniform-width anisotropic porous channel. The model incorporates a generalized Brinkman-Darcy formulation for the porous layers while considering the motion of nanoparticles influenced by both Brownian diffusion and thermophoresis effects.Design/methodology/approachThe similarity transformations derived through Lie group analysis are used to reduce the system from nonlinear partial differential equations to nonlinear ordinary differential equations. The finite difference method-based numerical routine bvp4c is employed to collect and graphically present the outcomes for velocity, temperature, and nanoparticle concentration profiles. The flow pattern is analyzed through streamlined plots. Furthermore, skin friction, heat, and mass transmission rates are investigated and presented via line plots.FindingsIt is observed that in anisotropic porous media, the temperature profile is stronger than in isotropic porous media. The thermal anisotropic parameter enhances the concentration profile while reducing the temperature.Practical implicationsAnisotropy arises in various industrial and natural systems due to factors such as preferred orientation or asymmetric geometry of fibers or grains. Hence, this study has applications in oil extraction processes, certain fibrous and biological materials, geological formations, and dendritic zones formed during the solidification of binary alloys.Originality/value1. The permeability and thermal conductivity are not constant; instead, they have different values in the x and y directions. 2. This study considers the dependency of thermophoresis on nanoparticle volume fraction and Brownian diffusion on the temperature in both the fluid flow equations and boundary conditions. 3. A novel similarity transformation is derived using Lie group analysis instead of using an existing transformation already available in the literature.

On the resonance frequency of the irregular shaped visco-acoustic Helmholtz cavity problem

In this paper we develop a closed-form mathematical solution for the resonance frequency of the axisymmetric irregular shaped visco-acoustic Helmholtz cavity problem. The solution is based on a conformal mapping that transforms the governing visco-acoustic equations of motion for the irregular cavity geometry into a new domain described by a set of orthogonal curvilinear coordinates. These orthogonal curvilinear coordinates have an important characteristic relative to the transformation of the physical boundary: the mapped domain is constructed such that the locus of all points for the length-wise profile of the physical boundary coincides with a constant value for a (quasi radial) coordinate in the new mapped domain. In these orthogonal curvilinear coordinates we obtain an analytical solution for the visco-acoustic equations of motion and exactly satisfy the no-slip fluid boundary conditions over the irregular shaped cavity walls by matching the coefficients of an orthogonal Legendre polynomial series representation for the non-uniform velocity on the cavity walls. The mathematical solution is an approximation to the extent that the scale factors chosen for the mapping consist of strictly the mutual-coupling terms of the true scale factors. Thus we analyze some results to understand the correlation of the analytical solution frequency response spectra with numerical finite element analyses of the idealized boundary conditions with various irregular shaped cavity geometries. The results for fundamental resonance frequency were found to exhibit good correlation with the results of finite element analyses over a significant range of fluid properties with different irregular cavity geometries. As the primary motivation for the derivation stems from the lack of a rapid yet accurate alternative to computationally intensive MultiPhysics FEA in the formulation of acoustic trendlines for sensor development, the solution’s acoustic impedance was applied as dispersive loading in a simple lumped parameter model of a fluid identification sensor for comparison. The resulting acoustic trendlines of the reduced (5 DOF) model were found to exhibit good correlation with the results of the MultiPhysics FEA (>105 DOF). Thus the solution was found to enable a closed-form analytical method for rapid construction of parametric studies in the practical design of fluid analysis sensors based on complex resonant cavity geometries.

A new analytical model for transient heat transfer modeling of geothermal monotube heat exchangers

In this study, we present a new analytical model for the analysis of transient heat transfer (the model is solved after Laplace transformation) in geothermal monotube heat exchangers (GHE). The model accounts for various factors, including the heat exchanger's geometric parameters, the nature of the fluid (air and water), flow characteristics (laminar and turbulent), and heat transfer boundary conditions. Initially, the model's performance is validated by comparing its results with those obtained from computational fluid dynamics simulations using OpenFOAM. This validation process ensures the reliability and accuracy of the model. Additionally, an analysis of the heat transfer mechanisms and key modeling parameters identified from the comparison is provided. In the subsequent section, the analysis is extended by comparing the proposed model to four existing analytical models available in the literature. These reference models employ distinct approaches to simulate heat transfer to the ground. By examining a range of cases, the performance of the model is systematically evaluated against these conventional methods. This work closes by an analysis of the analytical models and the impact on the accuracy of the results highlighting the advantages of the proposed method in geothermal applications.

Surface water quality modelling with data scarcity in semi-enclosed coastal regions encompassed distributed islands

Water quality variation in semi-enclosed urban coastal areas with different pollutant sources is a substantial issue. Pollutant entrapment has a significant impact on the lives of the local people. Surface water quality modelling often requires large datasets covering bathymetry, fluid and pollutants boundary conditions, sources and sinks. This is particularly challenging in regions with complex features and poor data availability, such as coastal water bodies featuring a large number of widely distributed islands and estuaries. In this paper, we developed a model of surface water quality and hydrodynamics for Ha Long Bay, in the North of Vietnam and includes 1969 islands, for a one-year period using a water quality dataset obtained for this study. The model utilized extracted bathymetric and geometric data from Admiralty Charts and Admiralty Tide Tables. Water quality coupled with the TELEMAC Model (WAQTEL) Biomass Module was employed to predict pollutant transport in the domain using point and diffused sources while field studies were conducted to collect data for the setting up and calibration of the water quality model. Thirty scattered sampling points were selected, and the water quality parameters were measured during two campaigns. The predicted water level and velocity values matched the local observation data well with a small error (RMSE = 0.19 and 0.16). Both NO3−-N and PO43--P were high near the shoreline and decrease gradually offshore. The maximum concentrations of NO3−-N and PO43--P reached 0.476 mg/L and 0.048 mg/L at the end of 2021 with the RMSE = 0.13 and 0.011, respectively. The levels of NO3−-N and PO43--P and their distributions showed that Ha Long Bay was eutrophic even during the COVID-19 lockdown period.

Chemical reaction effect across the moving flat plate with heat generation and MHD flow of Maxwell fluid with viscous dissipation

In this paper, the investigation centered on examining a Maxwell fluid’s convective double diffusive flow carefully considers factors such as chemical reactions, radiation, and the presence of a permeable moving flat plate. Additionally, the study encompassed the effects of heat generation and magnetohydrodynamics (MHD). The governing equations, initially expressed as partial differential equations, were converted into ordinary differential equations using a similarity transformation technique to facilitate the analysis. The computational power of MATLAB’s BVP4C software was harnessed to solve this resultant system of ODEs efficiently. The outcomes of this investigation were presented in the form of graphical representations that vividly depicted the behavior of the flow field, energy conservation, and concentration profiles under various parameter combinations. The research findings were thoughtfully summarized in a table, offering a comprehensive overview of temperature, velocity, and mass profiles across various parameters. These parameters included the Deborah number, chemical reaction rate, Eckert number, Lewis number, Prandtl number, porosity parameter, and MHD parameter. A notable discovery emerging from this study was the inverse relationship observed between nondimensional concentration contours and the magnitude of the chemical reaction rate. Simultaneously, it was observed that higher values of the Maxwell fluid led to a rise in the thickness of the temperature boundary layer. These findings offer valuable insights into the intricate interplay of physical and chemical phenomena in convective flows involving complex fluid properties and boundary conditions. The temperature outline diminishes as the heat generation rate increases, while the concentration profile declines with an elevation in the chemical reaction rate.

The absorbing boundary conditions of Newtonian fluid flowing across a semi-infinite plate with different velocities and pressures

Abstract The Newtonian fluid flowing across a semi-infinite plate with variable velocity and pressure is considered in this work. The dimensionless governing equation is obtained by introducing the dimensionless quantities. For infinite region, the artificial boundary approach by using the Laplace transform is applied to gain the absorbing boundary condition (ABC) in a finite region which we call the inner region. The approach differs from the traditional approximation method for infinite boundaries with large values and is first applied to the research. And the stability of the ABC is verified by considering the same point of the outer region and inner region. The numerical difference scheme by using the L1-scheme to approximate the fractional derivative is used to get solutions, and the feasibility assessments, such as stability and convergence, are developed. Three numerical examples are given. In the first example, the exact solution is gained by importing a source term. Through the comparison of numerical solution with exact solution verifies the accuracy of difference method. A comparison between the velocity distribution of the ABC and the infinite boundary approximated by a large value is also discussed and graphically analyzed. In the following two examples, by analyzing the fluid flow over the plate with assorted speeds or pressure gradient, the impact of correlative parameters on the velocity distribution and the flow mechanism are thoroughly analyzed.

Nonlinear geometric fluid-structure interaction model of multilayered sandwich plates in contact with unbounded or bounded fluid flow

Based on the Kirchhoff-Love theory and von Kármán's geometric nonlinearity, the nonlinear vibration analysis of multilayered sandwich plates under ideal flow is presented. Rectangular plates are considered to be laminated of orthotropic composite layers as upper and lower face sheets as well as a central lightweight core made of functionally graded (FG), anisogrid lattice or metal foam materials. Three coupled equations of motion are reformulated into one decoupled governing equation in terms of transverse deflection (w) using the method of inverse differential operator. For the first time, an impermeability condition is developed that considers the moderately large deflection of the contact surface in order to correct the pressure distribution of fluid flow. Based on the potential flow theory, closed-form expressions of hydrodynamic pressure are obtained for different fluid boundary conditions including infinite depth of fluid, flow bounded by a rigid wall and flow between two identical elastic plates. Then, the Galerkin and multiple-scale perturbation methods are used to obtain closed-form solution of nonlinear natural frequency in terms of maximum vibration amplitude. Finally, case studies are presented to investigate the effects of geometric ratios, boundary condition, material distribution, layup, characteristics of flow and geometric nonlinearity on the response quantities. Findings indicate that the large deformation term in the developed impermeability condition has significant effect on the pressure distribution and its effect is more pronounced for thicker plates with larger mode number in the flow direction. Therefore, the novel nonlinear fluid-structure interaction model should be considered to study flow-induced dynamic behavior of sandwich plates with different boundary conditions. In addition, the snap-through behavior is seen in the nonlinear frequency-upstream speed curves and the incipient points of dynamic instability can vary remarkably with the fluid boundary conditions.

Core-scale numerical simulation and comparison of breakdown of shale and resulting fractures using sc-CO2 and water as injectants

Supercritical carbon dioxide (sc-CO2) is an alternative to water for stimulation of low permeability systems such as shale gas and geothermal resources. Previously core-scale experimental studies have compared the behavior of CO2 to water injection for sample breakdown. Due to differences in experimental setup and core sample preparation, inconsistent or even apparently contradictory conclusions have resulted. To reconcile this contradiction, a phase-field numerical model is applied to understand hydraulic fracturing experiments using Green River shale found in the literature. The finite element numerical model incorporates a rate-dependent phase-field fracture model developed separately to describe fracture initiation and growth. We investigate the impact of various material and fluid properties on the resulting fractures. Most importantly, we study the effect of fluid properties and boundary conditions on the breakdown pressure, including the direction of the resulting fracture plane. Model results predict that (1) sc-CO2 injection in the laboratory may result in greater breakdown pressure than that of water under no-flow boundary conditions because lower viscosity sc-CO2 may result in pressure build up at the core boundary that opposes fracture initiation and (2) lower viscosity sc-CO2 also produces fast-propagating fractures that are less influenced by the bedding plane on their resulting fracture topology. Our model offers a straightforward explanation and reconciliation of existing experimental observations, as well as a means to extrapolate to new conditions. Exploration of field-scale conditions suggests less pronounced or no elevation in breakdown pressure when sc-CO2 is injected because the pressure build up effect at the system boundary is significantly less or absent at field length scales.

Existence of a solution to the non-steady magnetohydrodynamics-Boussinesq system with mixed boundary conditions

In this paper we are concerned with the non-steady magnetohydrodynamics (MHD)-Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include the stick, pressure, vorticity and stress conditions together, and in the case of static pressure may include the friction type of boundary conditions more. For the magnetic field, a non-homogeneous mixed boundary conditions are given. The conditions for temperature may include Dirichlet, Neumann and Robin conditions together. For the problem involving the static pressure and stress boundary conditions for fluid it is proved that if the data of problem are small enough, material parameters are independent of temperature, then under compatibility conditions in the initial time there exists a strong solution and the solution with small norm is unique. Much effort is paid to overcome a difficulty in dealing the advection term, Lorentz force in the momentum equation of fluid and a nonlinear term in the magnetic equation under the static pressure boundary conditions on a portion of boundary where there is flux of fluid and non-homogeneous magnetic boundary conditions. For the problem involving the total pressure and total stress boundary conditions for fluid, the existence of a solution is proved without smallness of the data of problem.

Evanescent waves in hybrid poroelastic metamaterials with interface effects

The propagation of evanescent waves in hybrid poroelastic metamaterials is investigated by considering interface effects. Hybrid metamaterials consist of a single-phased (acoustic or elastic) medium and a poroelastic medium. To establish the finite element model of elastic/poroelastic and fluid/poroelastic interfaces, weak integral forms of wave equations for elastic, fluid, and poroelastic media and boundary conditions at the interfaces between different media are first given. Next, the expressions for displacement and pressure in the frame of Bloch’s theorem are substituted into the dynamical equations to obtain general forms suitable for periodic metamaterials, from which the complex band structure and the frequency response of hybrid metamaterials are calculated. The influence of geometrical and material parameters, as well as the viscosity of the pore fluid on the propagation of elastic waves are discussed. The results and discussions show that flat bands and narrow locally resonant band gaps appear for elastic/poroelastic metamaterials. A quasi-resonance Bragg band gap is formed in the case of an elastic inclusion in a poroelastic matrix. Furthermore, a transition from an avoided crossing to a wave-number band gap is obtained by adjusting the geometrical and material parameters of the elastic inclusion. For the case of fluid inclusion in a poroelastic matrix with the open-pore interface, a cut-off frequency for the fast longitudinal wave is observed. However, only an avoided crossing is produced for the sealed-pore interface. For both cases, the phase velocity of the shear vertical (SV) wave decreases faster than that of the slow longitudinal wave as the radius of the inclusion increases. When the viscosity of the pore fluid is considered for elastic inclusion in a poroelastic matrix, the transmission dip at the ‘quasi-resonance Bragg’ band gap disappears. However, the transmission dip in the locally resonant band gap of the SV wave becomes slightly shallower and smoother than in the inviscid case. This study is relevant to practical applications of hybrid single-phased and poroelastic metamaterials, e.g., for coastal engineering and civil engineering.

Investigating the effect of wall temperature on the thermal performance of NH3 in the presence of suspended roughness in the 2D and 3D nanochannels: A molecular dynamics study

In thermal channels, the heat transfer coefficient increases by variations the fluid conditions, boundary conditions, or wall temperature. This article investigated the ammonia fluid manner with suspended roughness, in a nanochannel via molecular dynamics (MD) approach and LAMMPS open-source software. The change in thermal conductivity and heat flux were investigated to investigate the nanostructure's thermal behaviour. And its atomic behaviour in the nanochannel was investigated by examining the density and velocity profiles. In this study, two 2D and 3D copper nanochannels have been used. At first step, the equilibrium in the simulated structure was checked by examining the kinetic energy and potential changes. The results show that the peak point of fluid density in the areas near the wall of the nanochannel is higher than the middle areas in the nanochannel. This is because the intermolecular interaction is greater in the fluid molecules near the wall and eventually becomes smaller and more uniform in the middle regions of the fluid. Also, the results show that 2D and 3D nanochannel thermal conductivity converges to a numerical value of 0.53143 and 0.6063 W/mK after 10 ns. By changing the nanochannel wall's temperature, the motion of the particles in the channel increases. So the thermal conductivity in 2D and 3D nanochannel increase to 0.7176 and 0.7677 W/mK. The results will be effective in improving the performance of heating systems.

A Robin‐Neumann scheme with quasi‐Newton acceleration for partitioned fluid‐structure interaction

AbstractThe Dirichlet–Neumann scheme is the most common partitioned algorithm for fluid‐structure interaction (FSI) and offers high flexibility concerning the solvers employed for the two subproblems. Nevertheless, it is not without shortcomings: to begin with, the inherent added‐mass effect often destabilizes the numerical solution severely. Moreover, the Dirichlet–Neumann scheme cannot be applied to FSI problems in which an incompressible fluid is fully enclosed by Dirichlet boundaries, as it is incapable of satisfying the volume constraint. In the last decade, interface quasi‐Newton methods have proven to control the added‐mass effect and substantially speed up convergence by adding a Newton‐like update step to the Dirichlet–Neumann coupling. They are, however, without effect on the incompressibility dilemma. As an alternative, the Robin‐Neumann scheme generalizes the fluid's boundary condition to a Robin condition by including the Cauchy stresses. While this modification in fact successfully tackles both drawbacks of the Dirichlet–Neumann approach, the price to be paid is a strong dependency on the Robin weighting parameter, with very limited a priori knowledge about good choices. This work proposes a strategy to merge these two ideas and benefit from their combined strengths. The resulting quasi‐Newton‐accelerated Robin‐Neumann scheme is compared to both Robin‐ and Dirichlet–Neumann variants. The numerical tests demonstrate that it does not only provide faster convergence, but also massively reduces the influence of the Robin parameter, mitigating the main drawback of the Robin‐Neumann algorithm.

Morphodynamics of vortex ripple creation under constant and changing oscillatory flow conditions

The morphodynamics of orbital vortex ripples due to oscillatory flow and induced sediment transport was studied numerically. The methodology was based on large-eddy simulations of the full coupling of turbulent oscillatory fluid flow, bed and suspended sediment transport, and bed evolution. The Immersed Boundary method was implemented for the imposition of fluid and sediment boundary conditions on the mobile bed surface. Results are presented for ripple creation starting from flat or rippled beds, as well as results of ripples adapting to changing flow (wave) direction. The computed equilibrium ripple dimensions are in agreement with existing empirical predictions; however, the computed ripple profile is not as steep in the vicinity of the ripple crest as the analytical one in Sleath (1984). The time needed for the bed geometry to reach equilibrium, in cases where the initial bed is almost flat or it is formed by ripples that shorten towards equilibrium, is longer than in cases where the initial bed is formed by ripples that widen towards equilibrium. Moreover, the ripple crests are reoriented when the flow changes direction by 45°; the new equilibrium ripple crestline is perpendicular to the flow direction. Finally, the numerical results are compared with corresponding results of flow above fixed ripples. The bed load flux does not present substantial differences between the fixed and mobile bed cases. On the contrary, a strong increase of the suspended load flux is observed for the mobile bed case in comparison to the fixed bed case. • LES Coupling between oscillatory flow, sediment transport, and bed morphodynamics. • Same flow forcing results into same bed equilibrium regardless of initial bed form. • Evolution of ripples longer than equilibrium ones is slower than for shorter ones. • Ripples are reoriented perpendicular to the flow direction. • Suspended load over mobile rippled bed is larger than over fixed ripples.

Effect of fluid properties and boundary conditions on rod pump efficiency – Novel modeling

Among artificial lift systems, the most widely used method is sucker rod pumping. Since its inception in the oil industry, inefficiency in sucker rod pumping system due to gas interference is one of major concerns. The need for better understanding of gas interference during sucker rod pumping process is significantly required. Although, many models and techniques are available but there are many parameters, which need to be well studied and adequately addressed. This delineates the need to research all empirical parameters that could affect efficiency of conventional rod pump. The pump performance appears to be low in presence of gas which in extreme situation locks the pump and make the pump paralyzed completely. The models available in literature neglect mass transfer during suction process, which brings unreliable calculation of gas phase in pump. Models in literature on efficiency of pump and gas lock are studied to analyze the vital parameters and reasons for the inefficiency. The expedient analytical models to predict gas lock criteria and efficiency of conventional pump, based on liberation of gas during pumping, are established in this work. The models emphasize the importance of the fluids properties to be handled by pump, which affect the production rate while pumping with a gaseous phase. This new model considers free gas phase presence and mass transfer during working of pump. Sensitivity and comparative analysis of numerous parameters contributing to pump efficiency is established in this work. The study results provide some theoretical and analytical references for further advancements in system efficiency of conventional rod pump system and to improve economic benefits.

Modelling fluid flow in carbon fibre porous media based on X-ray microtomography and lattice Boltzmann method

Carbon fibre media possess complex internal porous structures; the microstructures are vital and can impact the efficiency of fluid flow and mass transfer. In this study, X-ray microtomography (XMT) and the lattice Boltzmann method (LBM) was employed to reconstruct the authentic three-dimensional (3D) pore structures of carbon fibre media and simulate its fluid flow. The effects of fluid properties and boundary conditions were elucidated. Results demonstrated that the flow velocity increased as the displacement differential pressure increased, but decreased as the fluid kinematic viscosity increased. The critical values of differential pressure and viscosity were 4.0106 Pa and 0.0101 m2/s, respectively. The simulation results of the average volume velocity revealed that three boundary conditions had extremely small relative errors of 0.0037 % depending on the carbon fibre material. When the Kozeny–Carman constant (KC, k) was 3.5, the relative error of permeability between the predicted values and the simulated values decreased with increasing porosity. When the porosity was 54.069 %, the relative error decreased to 4.18 %. The fractal model revealed worse results for the permeability prediction of carbon fibre porous media, and the relative error between the simulated and predicted values was 50.40 % when the porosity was 45.585 %.

Multiphysics Design of an Automotive Regenerative Eddy Current Damper

This research presents a finite element multi-physics design methodology that can be used to develop and optimise the inherent functions and geometry of an innovative regenerative eddy current (REC) damper for the suspension of B class vehicles. This methodology was inspired by a previous work which has been applied successfully for the development of an eddy current (EC) damper used for the same type of applications. It is based on a multifield finite element coupled model that can be used to fulfil the electromagnetic, thermal, and fluid dynamic field properties and boundary conditions of a REC damper, as well as its non-linear material properties and boundary conditions, while also analysing its damping performance. The proposed REC damper features a variable fail-safe damping force, while electric power is advantageously regenerated at high suspension frequencies. Its damping performance has been benchmarked against that of a regular hydraulic shock absorber (selected as a reference) by analysing the dynamic behaviour of both systems using a quarter car suspension model. The results are expressed in terms of damping force, harvested power, thermal field, comfort and handling, with reference to ISO-class roads. The optimisation analysis of the REC damper has suggested useful guidelines for the harmonisation of damping and regenerative power performances during service operation at different piston speeds.

Pulsatile parallel flow of air and a viscoelastic fluid with multiple characteristic times. An application to mucus in the trachea and the frequency of cough

We study the dynamics of a binary fluid, where the two fluids are flowing parallel to each other in a cylindrical geometry, and driven by a pulsatile pressure gradient. One of the fluids is a low viscosity Newtonian fluid, the other one is viscoelastic. In order to be able to apply the model to different biofluids, we consider that the viscoelastic fluid has several characteristic times. We characterize the dynamics of the fluids as generalized Darcy’s laws, with linear response functions to pulsatile pressure gradients, whose parameters are coupled for both fluids through the fluid–fluid boundary conditions. We apply our results to the dynamics of mucus and air in the trachea and find that the frequency that allows for a larger movement of the mucus, coincides with the experimental frequency of cough. This allows us to propose a plausible explanation for the frequency of cough in healthy individuals, a mechanical process to expel noxious substances from the respiratory system.

Existence of a solution to the steady Magnetohydrodynamics‐Boussinesq system with mixed boundary conditions

In this paper, we are concerned with the steady magnetohydrodynamics (MHD)‐Boussinesq system with mixed boundary conditions. The boundary conditions for fluid may include the stick, Tresca slip, leak, one‐sided leaks, pressure, vorticity, and stress conditions together, and the conditions for temperature may include Dirichlet, Neumann, and Robin conditions together. For the electromagnetic field, mixed boundary conditions are given. Also, the viscosity, magnetic permeability, electrical conductivity, thermal conductivity, and specific heat of the fluid depend on the temperature. Special attention is paid to the meaning of non‐homogeneous mixed boundary conditions for magnetic field. For the problem involving the static pressure and stress boundary conditions for fluid, it is proven that if the data of problem are small enough, then there exists a solution and the solution with small norm is unique when the viscosity, magnetic permeability, electrical conductivity, thermal conductivity, and specific heat are independent of the temperature. For the problem involving the total pressure and total stress boundary conditions for fluid, the existence of a solution is proven when a datum concerned with non‐homogeneous boundary conditions for magnetic field is small, but without smallness of other data.

An updated Gappy-POD to capture non-parameterized geometrical variation in fluid dynamics problems

In this work, we propose a new method to fill the gap within an incomplete turbulent and incompressible data field in such a way to satisfy the topological and intensity changes of the fluid flow after a non-parameterized geometrical variation in the fluid domain. This work extends the one that has been published as a conference proceeding to the 2018 AIAA Scitech Forum and Exposition (Akkari et al. in Geometrical reduced order modeling (ROM) by proper orthogonal decomposition (POD) for the incompressible navier stokes equations. In: 2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2018-1827), 2018). A single baseline large eddy simulation (LES) is assumed to be performed prior geometrical variations. The proposed method is an enhancement of the Gappy-POD method proposed by Everson and Sirovich in 1995, in the case where the given set of empirical eigenfunctions is not sufficient and is not interpolant for the recovering of the modal coefficients for each Gappy snapshot by a least squares procedure. This happens when the available data cannot be written as an interpolation of the baseline POD modes. This is typically the case when we introduce non-parameterized geometrical modifications in the fluid domain. Here, after the baseline simulation, additional solutions of the incompressible Navier–Stokes equations are solely performed over a restricted fluid domain, that contains the geometrical modifications. These local LESs that we will call hybrid simulations are performed by using the immersed boundary technique, which uses of a fluid boundary condition and the baseline velocity field. Then, we propose to update the POD modes using a local modification of the baseline POD modes in the restricted fluid domain. Furthermore, we will propose a physical correction of the latter enhanced Gappy-POD modal coefficients thanks to a Galerkin projection of the Navier–Stokes equations upon the new modes of the available data. This enhancement procedure on the global velocity reconstruction by the physical constraint was tested on a 3D semi-industrial test case of a typical aeronautical injection system and, a 2D laminar and unsteady incompressible test case. The speed-up relative to this new technique is equal to 100, which allows us to perform an exploration of two new designs of the aeronautical injection system.