Incompressibility Condition Research Articles

Analysis of the Staggered DG Method for the Quasi-Newtonian Stokes flows

AbstractThis paper introduces and analyzes a staggered discontinuous Galerkin (DG) method for quasi-Newtonian Stokes flow problems on polytopal meshes. The method introduces the flux and tensor gradient of the velocity as additional unknowns and eliminates the pressure variable via the incompressibility condition. Thanks to the subtle construction of the finite element spaces used in our staggered DG method, no additional numerical flux or stabilization terms are needed. Based on the abstract theory for the non-linear twofold saddle point problems, we prove the well-posedness of our scheme. A priori error analysis for all the involved unknowns is also provided. In addition, the proposed scheme can be hybridizable and the global problem only involves the trace variables, rendering the method computationally attractive. Finally, several numerical experiments are carried out to illustrate the performance of our scheme.

Non-equilibrium turbulent boundary layers in high reynolds number flow at incompressible conditions: effects of streamline curvature and three dimensionality

The physics and computational prediction of turbulent boundary layer flow over axisymmetric and three-dimensional bodies are examined. Three cases were considered for which extensive experimental results and companion Reynolds-averaged Navier Stokes (RANS) solutions were obtained and/or available in the open literature. These cases all have Reynolds numbers based upon the freestream velocity and body geometric scale on the order of 10 5 to 10 6 , which is large for laboratory scales but small compared to the maximum scales observed for full-scale vehicles. Despite significant differences in approach flow fields and geometries for these three cases, some common themes emerged in the findings. All cases involved complications due to pressure gradients combined with streamwise curvature, and all exhibited regions of turbulence reduction due to accelerated flow. These complications led to discrepancies in computed results even in attached flow regions where it is often assumed that RANS models provide reliable predictions. The authors recommend further work on modelling approaches that can capture rapid distortion effects on turbulence transport that can be incorporated into industry-useful frameworks. Two cases involving laterally symmetric, three-dimensional wall-mounted hills with aft-body separation revealed that asymmetric mean flow fields are likely to result. This finding has been observed in experiments conducted in multiple facilities and in computations using multiple solvers and turbulence models. It is concluded that non-unique and asymmetric global flow solutions are fundamental to flow cases with lateral geometric symmetry involving turbulent boundary layer separation. Further work is also needed to accurately predict low-frequency unsteadiness due to geometries that produce non-unique mean flow fields. For such flows, it remains to be definitively determined whether experimentally observed modes of the mean flow are equivalent, or nearly equivalent, to asymmetric mean flow solutions obtained using RANS approaches.

Odd viscous flow past a sphere at low but non-zero Reynolds numbers

The measurement of lift on symmetrically shaped obstacles immersed in low Reynolds number flow is the quintessential way to signal odd viscosity. For flow past cylinders, such a lift force does not arise if incompressibility and no-slip boundary conditions are fulfilled, whereas for spheres, a lift force has been found in Stokes flow, which is valid for cases where the Reynolds numbers are negligible and convection can be ignored. When considering the role of convection at low but non-zero Reynolds numbers, two hurdles arise, the Whitehead paradox and the breaking of axial symmetry, which are overcome by the method of matched asymptotic expansions and the Lorentz reciprocal theorem, respectively. We also consider the case where axial symmetry is preserved because the translation of the sphere is aligned with the axis of chirality of odd viscosity. We find that while lift vanishes, the interplay between odd viscosity and convection gives rise to a stream-induced torque.

Divergence-free cut finite element methods for Stokes flow

We develop two unfitted finite element methods for the Stokes equations based on Hdiv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^{{{\\,\ extrm{div}\\,}}}$$\\end{document}-conforming finite elements. Both cut finite element methods exhibit optimal convergence order for the velocity, pointwise divergence-free velocity fields, and well-posed linear systems, independently of the position of the boundary relative to the computational mesh. The first method is a cut finite element discretization of the Stokes equations based on the Brezzi–Douglas–Marini (BDM) elements and involves interior penalty terms to enforce tangential continuity of the velocity at interior edges in the mesh. The second method is a cut finite element discretization of a three-field formulation of the Stokes problem involving the vorticity, velocity, and pressure and uses the Raviart–Thomas (RT) space for the velocity. We present mixed ghost penalty stabilization terms for both methods so that the resulting discrete problems are stable and the divergence-free property of the Hdiv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{H}^{{{\\,\ extrm{div}\\,}}}$$\\end{document}-conforming elements is preserved also on unfitted meshes. In both methods boundary conditions are imposed weakly. We show that imposing Dirichlet boundary conditions weakly introduces additional challenges; (1) The divergence-free property of the RT and the BDM finite elements may be lost depending on how the normal component of the velocity field at the boundary is imposed. (2) Pressure robustness is affected by how well the boundary condition is satisfied and may not hold even if the incompressibility condition holds pointwise. We study two approaches of weakly imposing the normal component of the velocity at the boundary; we either use a penalty parameter and Nitsche’s method or a Lagrange multiplier method. We show that appropriate conditions on the velocity space has to be imposed when Nitsche’s method or penalty is used. Pressure robustness can hold with both approaches by reducing the error at the boundary but the price we pay is seen in the condition numbers of the resulting linear systems, independent of if the mesh is fitted or unfitted to the boundary.

Nonlinear twofold saddle point-based mixed finite element methods for a regularized μ(I)-rheology model of granular materials

We propose and analyze new mixed finite element methods for a regularized μ(I)-rheology model of granular flows with an equivalent viscosity depending nonlinearly on the pressure and the Euclidean norm of the symmetric part of the velocity gradient. To this end, and besides the velocity, the pressure and the aforementioned strain rate, we introduce a modified stress tensor that includes the convective term, and the skew-symmetric vorticity, as auxiliary tensor unknowns, thus yielding a mixed variational formulation within a Banach spaces framework. Then, the pressure is obtained through an iterative postprocess suggested by the incompressibility condition of the fluid, which allows us to express this unknown in terms of the aforementioned stress and the velocity. A fixed-point strategy combined with a solvability result for a class of nonlinear twofold saddle point operator equations in Banach spaces, is employed to show, along with the classical Banach fixed-point theorem, the well-posedness of the continuous and discrete formulations. In particular, PEERS and AFW elements of order ℓ⩾0 for the stress, the velocity, and the skew-symmetric vorticity, and piecewise polynomials of degree ⩽ℓ+n (resp. ⩽ℓ+1) for the strain rate with PEERS (resp. with AFW), yield stable Galerkin schemes. Optimal a priori error estimates are derived and associated rates of convergence are established. Finally, numerical results confirming the latter and illustrating the good performance of the methods, are reported.

Three-dimensional vorticity–velocity formulation in a lattice Boltzmann method

In recent decades, a paradigm shift in macroscopic methods has favored the use of non-primitive variables, such as velocity and vorticity (V–V), over traditional primitive variables. This shift eliminates the need for solving a Poisson equation for pressure, aligning numerical treatments more closely with physical reality. However, the lattice Boltzmann method (LBM), renowned for its efficacy in studying fluid flow phenomena, continues to rely on the conventional pressure–velocity (P–V) approach. This conventional approach necessitates a pressure–density relation, posing challenges in maintaining the incompressible condition. This study pioneers a novel application of the LBM to three-dimensional velocity–vorticity equations, expanding upon our suggested recent method for two-dimensional cases [Kefayati, Phys. Fluids. 36, 013128 (2024)]. To address the complexities introduced by the vortex stretching term in three dimensions, a new equilibrium distribution function is formulated and introduced to the three-dimensional nature of the vorticity vector. The paper details the derivation of the three-dimensional LBM and substantiates its effectiveness through numerical examples, showcasing its applicability in fluid dynamics. By bridging the gap between traditional P–V formulations and the benefits of non-primitive V–V variables, this work contributes to the ongoing exploration of LBM applications in fluid dynamics. The focus on three-dimensional scenarios involving velocity–vorticity equations marks a significant advancement, offering insights into the nuanced dynamics of fluid flow and paving the way for more accurate and realistic simulations in complex environments.

Determination of Local Heat Transfer Coefficients and Friction Factors at Variable Temperature and Velocity Boundary Conditions for Complex Flows

Transient conjugate heat transfer measurements under varying temperature and velocity inlet boundary conditions at incompressible flow conditions were performed for flat plate and ribbed channel geometries. Therefrom, local adiabatic wall temperatures and heat transfer coefficients were determined. The data were analyzed using typical heat transfer correlations, e.g., Nu=CRemPrn, determining the local distributions of C and m. It is shown that they are closely linked. A relationship lnC=A−mB is observed, with A and B as modeling parameters. They could be related to parameters in log-law or power-law representations for turbulent boundary layer flows. The parameter m is shown to have a close link to local pressure gradients and, therewith, near wall streamlines as well as friction factor distributions. A normalization of the C parameter allows one to derive a Reynolds analogy factor and, therefrom, local wall shear stresses.

An effective simulation scheme for the prediction of aerodynamic environment under hypersonic conditions characterized by NACA0012

Currently, aerodynamic environment prediction research into scramjet-propelled vehicles characterized by NACA0012 under hypersonic conditions is relatively sparse. Two-dimensional external flow field models are established, and then through validation tests, we perform a systematic investigation between simulation parameters and prediction accuracy, and an effective aerodynamic environment prediction simulation scheme under hypersonic conditions is proposed. Unlike under incompressible conditions, the maximum accuracy decline could be attributed to the inappropriate choice of the sharp trailing edge modeling method, but the definition formula is still preferred. In particular, for the two modeling data point sources, Airfoil tools and NACA4, the numerical performance of the latter is better than the former, and the calculation accuracy negatively correlates with the number of data points offered by both of them. Moreover, for the mesh cells near the shock, the cell Reynolds number and aspect ratio values should be no smaller than 16 and not exceed 380, respectively, and the recommended values for the far field distance, the turbulence model and flux type are 16L, Spalart-Allmaras, and ROE flux type. Under hypersonic conditions, the aerodynamic environment characterized by NACA0012 predicts a maximum temperature of approximately 1856.85 °C, with an average temperature change rate of 77 °C/s. Meanwhile, the top sound pressure level and the vibration acceleration could reach up to 145 dB and 182 g, respectively.

Finite element software for forming processes of glass containers

AbstractA two‐dimensional (2D) axisymmetric numerical model, based on the finite element method, for glass containers forming processes is presented. Glass forming processes involve coupled thermomechanical phenomena in which heat transfer and viscous flow are dependent, as glass viscosity is highly dependent on temperature. During the overall process glass changes from a molten state to a solid state. Therefore, adequate cooling conditions must be set appropriately. From the numerical point of view, the modeling must be robust so as to adjust to the different sequenced stages. Remeshing techniques requiring adequate data transfer, as well as, different thermal and mechanical contact conditions between glass and molds must be taken into account. Also, effective treatment of the incompressible conditions associated with glass flow must be dealt with. The aim is to set the better process parameters so that the final containers have the required geometrical shape and thickness distribution. A numerical model was conducted addressing all these issues and a thickness distribution comparison with real industrial products was performed.

Magnetic field effects on double diffusion of four sloshing fins in a nanofluid-occupied cavity

ABSTRACT This article offers a novel study on the impacts of magnetic field and Soret-Dufour numbers on the thermosolutal convection of sloshing four fins on the wall sides of a square cavity occupied by a nanofluid. The incompressible condition of smoothed particle hydrodynamics (SPH) entitled ISPH method simulated sloshing four fins inside a nanofluid flow contained by a square cavity. The scales of pertinent parameters are fin length 0.05 ≤ L Fin ≤ 0.3 , Dufour number 0 ≤ Du ≤ 1 , a magnetic field inclination angle 30 ∘ ≤ γ ≤ 90 ∘ , Soret number 0 ≤ Sr ≤ 2 , Hartmann number 10 ≤ Ha ≤ 50 , Rayleigh number 10 4 ≤ Ra ≤ 10 7 , and nanoparticle parameter 0 ≤ ϕ ≤ 0.2 . The simulations of the ISPH method indicated the significant role of the fin’s length in enhancing the nanofluid alterations and transmission of thermal/solutal convection inside a cavity. Besides, the Rayleigh number has a great influence on improving nanofluid velocity and power of temperature/concentration. The Soret/Dufour numbers are successful in boosting the nanofluid speed as well as heat and mass transport, while the Hartmann number and nanoparticles parameter are effective in reducing the nanofluid movements.

A partially debonded rigid elliptical inclusion with a liquid slit inclusion occupying the debonded portion

We derive a closed-form solution to the plane strain problem of a partially debonded rigid elliptical inclusion in which the debonded portion is filled with a liquid slit inclusion when the infinite isotropic elastic matrix is subjected to uniform remote in-plane stresses. The original boundary value problem is reduced to a Riemann–Hilbert problem with discontinuous coefficients, and its analytical solution is derived. By imposing the incompressibility condition of the liquid slit inclusion and balance of moment on a circular disk of infinite radius, we obtain a set of two coupled linear algebraic equations for the two unknowns characterizing the internal uniform hydrostatic tension within the liquid slit inclusion and the rigid body rotation of the rigid elliptical inclusion. As a result, these two unknowns can be uniquely determined revealing the elastic field in the matrix.

Investigation on algorithms for simulating large deformation and impact loads

It is a challenge to simulate the hydrodynamic problems covering the large deformation of the free surface arising in severe circumstances with intense flow. This paper investigates algorithms based on the moving particle semi-implicit method for simulating large deformation and impact loads. The algorithm discretizes the fluid domain into a series of particles, each representing a part of the fluid. The pressure field calculation is implicit, and the velocity field calculation is explicit. Three models, including the gradient model, source term, and free-surface detection, have been improved and compared to determine which improvement is the best to enhance the accuracy and stability. The enhanced pressure gradient guarantees that momentum conservation can be satisfied. Particle density and velocity divergence are incompressible conditions combined in the mixed source term approach. The arc approach is used in the free-surface judging process. The results show that the combination of three models is the most effective in exploring the problems of hydrodynamic pressure and dam break. The issue of liquid sloshing including roll and sway investigates the effect of the initial distance and time step. It is found that the simulation accuracy of impact pressure can be increased as the initial distance and the time step decrease. Finally, the free surface breaking and liquid splashing phenomena are easily observed, and the method can accurately simulate the massive deformation of the free surface. These findings are helpful for hazard assessments of the various fluid mechanics-related problems.

Rindler fluids from gravitational shockwaves

We study a correspondence between gravitational shockwave geometry and its fluid description near a Rindler horizon in Minkowski spacetime. Utilizing the Petrov classification that describes algebraic symmetries for Lorentzian spaces, we establish an explicit mapping between a potential fluid and the shockwave metric perturbation, where the Einstein equation for the shockwave geometry is equivalent to the incompressibility condition of the fluid, augmented by a shockwave source. Then we consider an Ansatz of a stochastic quantum source for the potential fluid, which has the physical interpretation of shockwaves created by vacuum energy fluctuations. Under such circumstance, the Einstein equation, or equivalently, the incompressibility condition for the fluid, becomes a stochastic differential equation. By smearing the quantum source on a stretched horizon in a Lorentz invariant manner with a Planckian width (similarly to the membrane paradigm), we integrate fluctuations near the Rindler horizon to find an accumulated effect of the variance in the round-trip time of a photon traversing the horizon of a causal diamond.

Study of nanofluid flow in a stationary cone–disk system with temperature-dependent viscosity and thermal conductivity

The substantial temperature gradient experienced by systems operating at relatively high temperatures significantly impacts the transport characteristics of fluids. Hence, considering temperature-dependent fluid properties is critical for obtaining realistic prediction of fluid behavior and optimizing system performance. The current study focuses on the flow of nanofluids in a stationary cone–disk system (SCDS), taking into account temperature-dependent thermal conductivity and viscosity. The influence of Brownian motion, thermophoresis, and Rosseland radiative flux on the heat transport features are also examined. The Reynolds model for viscosity and Chiam's model for thermal conductivity are employed. The Navier–Stokes equation, the energy equation, the incompressibility condition, and the continuity equation for nanoparticles constitute the governing system. The Lie-group transformations lead the self-similar ordinary differential equations, which are then solved numerically. Multi-variate non-linear regression models for the rate of heat and mass transfers on the disk surface were developed. Our study reveals a notable decrease in the rate of heat and mass transfer when pre-swirl exists in the flow. The significant influence of nanofluid slip mechanisms on the effective temperature and nanofluid volume fraction (NVF) within the system is highlighted. Furthermore, the variable viscosity property enhances the temperature and NVF of the SCDS.

Study of hybrid nanofluid flow in a stationary cone-disk system with temperature-dependent fluid properties

Cone-disk systems find frequent use such as conical diffusers, medical devices, various rheometric, and viscosimetry applications. In this study, we investigate the three-dimensional flow of a water-based Ag-MgO hybrid nanofluid in a static cone-disk system while considering temperature-dependent fluid properties. How the variable fluid properties affect the dynamics and heat transfer features is studied by Reynolds’s linearized model for variable viscosity and Chiam’s model for variable thermal conductivity. The single-phase nanofluid model is utilized to describe convective heat transfer in hybrid nanofluids, incorporating the experimental data. This model is developed as a coupled system of convective-diffusion equations, encompassing the conservation of momentum and the conservation of thermal energy, in conjunction with an incompressibility condition. A self-similar model is developed by the Lie-group scaling transformations, and the subsequent self-similar equations are then solved numerically. The influence of variable fluid parameters on both swirling and non-swirling flow cases is analyzed. Additionally, the Nusselt number for the disk surface is calculated. It is found that an increase in the temperature-dependent viscosity parameter enhances heat transfer characteristics in the static cone-disk system, while the thermal conductivity parameter has the opposite effect.

Low-Order Moments of Velocity Gradient Tensors in Two-Dimensional Isotropic Turbulence

In isotropic turbulence, symmetry of different directions can reduce the number of independent components for velocity gradient tensors. In three-dimensional isotropic turbulence, the independent components under either incompressible or compressible conditions have already been analyzed in the literature. However, for two-dimensional isotropic turbulence, they are still unclear. We derive rigorously the independent components for velocity gradient tensors of two-dimensional isotropic turbulence and give physical explanations. These theoretical results are validated using high-resolution direct numerical simulations (DNSs) of two-dimensional compressible turbulence. Results show that the present DNS setup is still not sufficient to capture the isotropy of third-order moments, suggesting that more investigations on determining the smallest scale and improving the numerical schemes for two-dimensional compressible turbulence are required.

Pressure stabilization method for the 2D/3D time‐dependent natural convection equations with salt concentration

AbstractIn this paper, the pressure stabilization method for the 2D/3D time‐dependent natural convection equations with salt concentration (NCSC) is considered. The artificial pressure diffusion term is introduced to reduce the constraint of the incompressible condition. The first‐order backward difference formula is adopted to approximate the temporal derivative terms, the diffusion terms in NCSC are treated implicitly, the convective terms in momentum equation are treated semi‐implicitly, the convective terms in energy equation and salt concentration equation are treated implicitly. The error estimates for the semi‐discrete pressure stabilization method are given. Some numerical experiments are presented to verify the theoretical results and show the reliability of the proposed method.

An incompressible liquid slit between dissimilar anisotropic elastic media

We use the Stroh sextic formalism to solve the generalized plane strain problem associated with a finite incompressible liquid slit lying between dissimilar anisotropic elastic half-planes subjected to a system of uniform remote stresses. The liquid slit admits an internal uniform hydrostatic stress field and is perfectly bonded to the surrounding media. By enforcing the interface conditions on the solid-solid and liquid-solid interfaces, a Riemann-Hilbert problem of vector form is obtained and is solved analytically using a decoupling procedure. An application of the residue theorem following the imposition of the incompressibility condition of the liquid slit leads to the determination of the internal uniform hydrostatic tension within the liquid slit. When the elastic bimaterial is orthotropic, the internal uniform hydrostatic tension is equal to the remote normal stress component perpendicular to the surfaces of the liquid slit. The presence of the liquid slit will inhibit crack growth along the anisotropic elastic bimaterial interface.

New non-augmented mixed finite element methods for the Navier–Stokes–Brinkman equations using Banach spaces

Abstract In this paper we consider the Navier–Stokes–Brinkman equations, which constitute one of the most common nonlinear models utilized to simulate viscous fluids through porous media, and propose and analyze a Banach spaces-based approach yielding new mixed finite element methods for its numerical solution. In addition to the velocity and pressure, the strain rate tensor, the vorticity, and the stress tensor are introduced as auxiliary unknowns, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation becomes a nonlinear perturbation of, in turn, a perturbed saddle point linear system, which is then rewritten as an equivalent fixed-point equation whose operator involved maps the velocity space into itself. The well-posedness of it is then analyzed by applying the classical Banach fixed point theorem, along with a smallness assumption on the data, the Babuška–Brezzi theory in Banach spaces, and a slight variant of a recently obtained solvability result for perturbed saddle point formulations in Banach spaces as well. The resulting Galerkin scheme is momentum-conservative. Its unique solvability is analyzed, under suitable hypotheses on the finite element subspaces, using a similar fixed-point strategy as in the continuous problem. A priori error estimates are rigorously derived, including also that for the pressure. We show that PEERS and AFW elements for the stress, the velocity, and the rotation, together with piecewise polynomials of a proper degree for the strain rate tensor, yield stable discrete schemes. Then, the approximation properties of these subspaces and the Céa estimate imply the respective rates of convergence. Finally, we include two and three dimensional numerical experiments that serve to corroborate the theoretical findings, and these tests illustrate the performance of the proposed mixed finite element methods.

Aerodynamic Analysis of a Logistics UAV with Foldable Bi-wing Configuration

Herein, a twin-boom, inverted V-tailed unmanned aerial vehicle (UAV) featuring a foldable bi-wing configuration is proposed for logistics and transportation applications. We employed the Navier–Stokes solver to numerically simulate steady, incompressible flow conditions. By examining the effects of key design parameters on aerodynamic characteristics and bypass flow fields in a two-dimensional state, we were able to suggest a more optimized foldable wing design. Building on the two-dimensional analysis, we performed aerodynamic assessments of the three-dimensional aircraft geometry. Our results indicated that appropriate wing and gap parameters can significantly enhance lift characteristics, maintaining high lift even during large-angle flights. Specifically, when compared to a mono-wing, the lift coefficient of the bi-wing increased by 27.1% at a 14° angle of attack, demonstrating the effectiveness of our wing-and-gap design. Optimal aerodynamic performance was achieved when the gap distance equalled the chord length in both flow and vertical directions. Further, the right combination of airfoil configuration, wing axes angle, and wingspan can improve flow field aerodynamic characteristics, while also enhancing the wing’s stall capacity. The lift coefficient reached its maximum value at an angle of attack of 15°, which has the potential to reduce takeoff and landing distances, thereby enhancing the UAV’s overall safety.