Incompressible Equations Research Articles

A reduced-order variational multiscale interpolating element free Galerkin technique based on proper orthogonal decomposition for solving Navier–Stokes equations coupled with a heat transfer equation: Nonstationary incompressible Boussinesq equations

In the recent decade, meshless methods have been handled for solving some PDEs due to their easiness. One of the most efficient meshless methods is the element free Galerkin (EFG) method. The test and trial functions of the EFG are based upon the special basis. Recently, some modifications have been developed to improve the EFG method. One of these improvements is the variational multiscale EFG (VMEFG) procedure. In the current article, the shape functions of interpolating moving least squares (IMLS) approximation are applied to the variational multiscale EFG technique to numerical study the Navier–Stokes equations coupled with a heat transfer equation such that this model is well-known as two-dimensional nonstationary Boussinesq equations. In order to reduce the computational time of simulation, we employ a reduced order model (ROM) based on the proper orthogonal decomposition (POD) technique. In the current paper, we developed a new reduced order model based on the meshless numerical procedure for solving an important model in fluid mechanics. To illustrate the reduction in CPU time as well as the efficiency of the proposed method, we investigate two-dimensional cases.

Global well‐posedness for the stochastic non‐Newtonian fluid equations and convergence to the Navier‐Stokes equations

We establish the existence of global pathwise solutions for the stochastic non‐Newtonian incompressible fluid equations in two space dimensions. Moreover, we show that said solutions converge in probability to solutions of the stochastic Navier‐Stokes equations in the appropriate limit. Our approach is based on Galerkin approximations and the theory of martingale solutions.

Entropy generation minimization of a pump running in reverse mode based on surrogate models and NSGA-II

Pumps in reverse function are one of the most cost-effective industrial devices to generate power from small hydropower capacities. Due to the fact that the pump components are not inherently designed for the turbine mode, the hydraulic performance of the pump as turbine (PAT) is usually not ideal. One low-cost solution to improve the PAT performance is to redesign the pump Impeller without changing of any other components. In this study a multi-objective optimization process is applied based on the surrogate models with focus on minimizing the entropy generation rate (EGR) at 2 flow rates of the QBEP (best efficiency point) and Q1.3BEP in order to improve the hydraulic performance of the PAT over a wide range of operating points. In the optimization process, the sample space points are determined by the optimal Latin hypercube sampling (OLHS) technique. The values of the objective functions and the constraints at the sample points are specified by numerical solution of the 3D incompressible steady-state RANS equations. Kriging (KRG) surrogate models have been constructed to estimate the objective functions and constraints at the whole sample space. The blade inlet and outlet angles, the blade wrap angle, the runner inlet width and the number of blades are defined as the design variables. Finally, by using non-dominated sorting genetic algorithm II (NSGA-II), in addition to obtaining the Pareto optimal fronts (POFs), the optimal combinations of the design variables are determined. Analysis of the importance level of the design variables has been performed using the RSA surrogate model. Optimization results reveals that the maximum reduction in the EGR occurs in the optimal 8-bladed runner by approximately 50%. The decrease in the entropy production leads to a decrease in irreversible hydraulic losses and an increase in the hydraulic efficiency of the PAT, e.g. 3.8% and 5.72% increase in efficiency at the QBEP and Q1.3BEP, respectively. Flow field analysis demonstrates that the entropy generation minimization causes a reduction in flow disorders within the optimal PATs. As a result, inlet shock, flow deviation at the blade outlet, flow separation at the blade passage, backflow and swirling flow at the draft tube are dramatically reduced or completely eliminated. The optimization of the PAT with the help of surrogate models and evolutionary algorithms (EAs), from the point of view of the second law of thermodynamics, has been successfully investigated in this work and the improvement of hydraulic performance was achieved.

Regularity criteria to the incompressible axisymmetric Boussinesq equations

In this paper, we are devoted to establishing some new regularity criteria of the weak solutions to the incompressible axisymmetric Boussinesq equations, which are independent of density. To this end, we establish some new a priori estimates. To some extent, our result can be thought as a generation of that in Chae and Lee (2002) and an addition of that in Fang et al. (2018).

Coarsening in the two-dimensional incompressible Toner-Tu equation: Signatures of turbulence.

We investigate coarsening dynamics in the two-dimensional, incompressible Toner-Tu equation. We show that coarsening proceeds via vortex merger events, and the dynamics crucially depend on the Reynolds number Re. For low Re, the coarsening process has similarities to Ginzburg-Landau dynamics. On the other hand, for high Re, coarsening shows signatures of turbulence. In particular, we show the presence of an enstrophy cascade from the intervortex separation scale to the dissipation scale.

On Marangoni shear convective flows of inhomogeneous viscous incompressible fluids in view of the Soret effect

A new exact solution is obtained for the Oberbeck-Boussinesq equations describing the steady-state layered (shear) Marangoni convection of a binary viscous incompressible fluid with the Soret effect. When layered (shear) flows are considered, the Oberbeck-Boussinesq system is overdetermined. For it to be solvable, a class of exact solutions is constructed, which allows one to satisfy identically the “superfluous” equation (the incompressibility equation). The found exact solution allows the Oberbeck-Boussinesq system of equations to be reduced to a system of ordinary differential equations by the generalized method of separation of variables. The resulting system of ordinary differential equations has an analytical solution, which is polynomial. The polynomial velocity field describes counterflows in the case of a convective fluid flow. It is demonstrated that the components of the velocity vector can have one stagnant (zero) point inside the region under study. In this case, the corresponding component of the velocity vector can be stratified into two zones, in which the fluid flows in opposite directions. The exact solution describing the velocity field for the Marangoni convection of a binary fluid has non-zero helicity, the flow itself being almost everywhere vortex.

Global solvability to the high-dimensional inhomogeneous Boussinesq equations with zero thermal diffusion

In this paper, we prove the global existence and uniqueness of the strong solution to the high-dimensional inhomogeneous incompressible Boussinesq equations with zero thermal diffusion.

Boundedness in a three-dimensional Keller–Segel–Stokes system with subcritical sensitivity

We consider a quasilinear Keller–Segel system with density-dependent migration rates, coupled to the incompressible Stokes equations through transport and buoyancy. By means of an apparently novel approach based on certain conditional estimates for the taxis gradient and the fluid field, for diffusion rates asymptotically controllable by power-tape majorants and minorants a result on global existence and boundedness is derived under an essentially optimal condition on the strength of cross-diffusion relative to diffusion.

Blow-up criterion for the density dependent inviscid Boussinesq equations

In this work, we consider the density-dependent incompressible inviscid Boussinesq equations in mathbb{R}^{N} (Ngeq 2). By using the basic energy method, we first give the a priori estimates of smooth solutions and then get a blow-up criterion. This shows that the maximum norm of the gradient velocity field controls the breakdown of smooth solutions of the density-dependent inviscid Boussinesq equations. Our result extends the known blow-up criteria.

An energy-stable scheme for a 2D simple fluid-particle interaction problem

We develop an energy-stable scheme for simulating fluid-particle interaction problems governed by a coupled system consisting of the incompressible Navier-Stokes (NS) equations defined in a time-dependent fluid domain and Newton's second law for particle motion. A modified temporary arbitrary Lagrangian-Eulerian (tALE) method is designed based on a bijective mapping between the fluid regions at different time steps. In the proposed numerical scheme, the tALE mesh velocity, the incompressible NS equations, and Newton's second law are solved simultaneously. We prove that under certain conditions, the new time discretization scheme satisfies an energy law. For the space discretization, the extended finite element method (XFEM) is used to solve the problem on a fixed Cartesian mesh. The developed method is first-order accurate in time and space without being momentum conservative. To verify the accuracy and stability of our numerical scheme, we present numerical experiments including the fitting of the Jeffery orbit by rotating of an ellipse and the free-falling of an elliptic particle in water.

Global Regularity of Three-Dimensional Incompressible Magneto-Micropolar Fluid Equations with Damping

We deal with the Cauchy problem of three-dimensional incompressible magneto-micropolar fluid equations with a nonlinear damping term $$\alpha |{\mathbf {u}}|^{\beta -1}{\mathbf {u}}\ (\alpha >0\ \text {and}\ \beta \ge 1)$$ in the momentum equations. By cancelation properties of the system under study, we show that there exists a unique global strong solution for any $$\beta \ge 4$$ . Our work extends previous results.

H(div) conforming methods for the rotation form of the incompressible fluid equations

New H(div) conforming finite element methods for incompressible flows are designed that involve the rotation form of the equations of motion and the Bernoulli function. With a specific choice of numerical fluxes, we recover the same velocity field as in Guzman et al. (IMA J Numer Anal 37(4):1733–1771, 2016) for the incompressible Euler equation in the convection form. Error estimates are presented for the semi-discrete method. We further study the incompressible Navier-Stokes equation with the full version of the stress tensor $$\nu \left( \nabla \varvec{u}+ \nabla \varvec{u}^T - \frac{2}{3} \left( \nabla \cdot \varvec{u}\right) \mathbb {I} \right)$$ , instead of partially enforcing the divergence free constraint at the continuous level (as is commonly done in finite element methods), we let the numerical scheme to fully control the enforcement of this constraint. Finally, we test the behavior of the proposed methods with some numerical simulations. Our results show that (1) We recover the same velocity field in Guzman et al. (2016), (2) When H(div) conforming with BDM-DG elements, we achieve less errors in the velocity compared with Schroeder et al. (SeMA J 75(4):629–653, 2018) when polynomial order $$p\in \{2,3\}$$ , (3) When H1 conforming with Taylor-Hood elements, the use of full stress tensor helps to reduce errors in both the velocity and the Bernoulli function, (4) H(div) conforming method does a better job in long time structure preservation compared with the classical mixed method even with the grad-div stabilization.

Turbulent bubbly channel flows: Effects of soluble surfactant and viscoelasticity

Interface-resolved direct numerical simulations are performed to examine the combined effects of soluble surfactant and viscoelasticity on the structure of a bubbly turbulent channel flow. The incompressible flow equations are solved fully coupled with the FENE-P viscoelastic model and the equations governing interfacial and bulk surfactant concentrations. The latter coupling is achieved through a non-linear equation of state which relates the surface tension to the surfactant concentration at the interface. The two-fluid Navier-Stokes equations are solved using a front-tracking method, augmented with a very efficient FFT-based pressure projection method that allows for massively parallel simulations of turbulent flows. It is found that, for the surfactant-free case, bubbles move toward the wall due to inertial lift force, resulting in formation of wall layers and a significant decrease in the flow rate. Conversely, a high-enough concentration of surfactant changes the direction of lateral migration of bubbles, i.e., the contaminated bubbles move toward the core region and spread out across the channel. When viscoelasticity is considered, viscoelastic stresses counteract the Marangoni stresses, promoting formation of bubbly wall-layers and consequently strong decrease in the flow rate. The formation of bubble wall-layers for combined case depends on the interplay of the inertial and elastic, and Marangoni forces.

Gravity dual of Navier-Stokes equation in a rotating frame through parallel transport

The fluid-gravity correspondence documents a precise mathematical map between a class of dynamical spacetime solutions of the Einstein field equations of gravity and the dynamics of its corresponding dual fluid flows governed by the Navier-Stokes (NS) equations of hydrodynamics. This striking connection has been explored in several dynamics-based approaches and has surfaced in various forms over the past four decades. In a recent construction, it has been shown that the manifold properties of geometric duals are in fact intimately connected to the dynamics of incompressible fluids, thus bypassing the conventional on-shell standpoints. Following such a prescription, we construct the geometrical description that effectively captures the dynamics of an incompressible NS fluid with respect to a uniformly rotating frame. We propose the gravitational dual(s) described by bulk metric(s) in $(p+2)$-dimensions such that the equations of parallel transport of an appropriately defined bulk velocity vector field when projected onto an induced timelike hypersurface require that the incompressible NS equation of a fluid relative to a uniformly rotating frame be satisfied at the relevant perturbative order in $(p+1)$-dimensions. We argue that free fluid flows on manifold(s) described by the proposed metric(s) can be effectively considered as an equivalent theory of non-relativistic viscous fluid dynamics with respect to (w.r.t) a uniform rotating frame. We also present suggestive insights as to how space-time rotation parameters encode information pertaining to the inertial effects in the corresponding fluid dual.

Cached Gaussian elimination for simulating Stokes flow on domains with repetitive geometry

Microfluidic “lab on a chip” devices are small devices that operate on small length scales on small volumes of fluid; these devices find uses in a variety of applications. Designs for microfluidic chips are generally composed of standardized and often repeated components connected by long, thin, straight fluid channels. We propose a novel meshing algorithm for use in simulating the linear incompressible stationary Stokes equations on geometry with these features, which produces sparse symmetric positive indefinite systems with many repeated matrix blocks. We use a discretization that is formally third order accurate for velocity and second order accurate for pressure in the L∞ norm. We also propose a novel linear system solver based on cyclic reduction, reordered sparse Gaussian elimination, and operation caching that is designed to efficiently solve systems with repeated matrix blocks. We demonstrate that the resulting fluid solver is significantly faster than existing methods up to resolutions of a few million degrees of freedom for microfluidic problems.

A high-order discontinuous Galerkin solver for the incompressible RANS equations coupled to the [formula omitted] turbulence model

Accurate methods to solve the Reynolds-Averaged Navier-Stokes (RANS) equations coupled to turbulence models are still of great interest, as this is often the only computationally feasible approach to simulate complex turbulent flows in large engineering applications. In this work, we present a novel discontinuous Galerkin (DG) solver for the RANS equations coupled to the k−ϵ model (in logarithmic form, to ensure positivity of the turbulence quantities). We investigate the possibility of modeling walls with a wall function approach in combination with DG. The solver features an algebraic pressure correction scheme to solve the coupled RANS system, implicit backward differentiation formulae for time discretization, and adopts the Symmetric Interior Penalty method and the Lax-Friedrichs flux to discretize diffusive and convective terms respectively. We pay special attention to the choice of polynomial order for any transported scalar quantity and show it has to be the same as the pressure order to avoid numerical instability. A manufactured solution is used to verify that the solution converges with the expected order of accuracy in space and time. We then simulate a stationary flow over a backward-facing step and a Von Kármán vortex street in the wake of a square cylinder to validate our approach.

Local Existence and Uniqueness of Strong Solutions to the Two Dimensional Nonhomogeneous Incompressible Primitive Equations

In this article, we study the initial boundary value problem of the two-dimensional nonhomogeneous incompressible primitive equations and obtain the local existence and uniqueness of strong solutions. The initial vacuum is allowed.

Application of the method of current functions in stationary plastic deformation processes

A method of determining the components of the strain rate tensor based on the method of current functions is developed. It is assumed that, with an axisymmetric plastic deformation of the metal in a channel with curvilinear borders, the kinematics of the process are similar to a flat flow. Using the incompressibility equation and the differential equation of the current lines, it is established that the current function maintains a constant value along the current lines. Two infinitely close current lines are considered in the flow plane to explain the kinematic content of the current function. The expression for flow through the finite transverse-sized tube was obtained. In the absence of radial velocity components at the boundaries, we obtain the restrictions imposed on derivatives of the current functions at these boundaries. The method of calculation of kinematic deformation characteristics for established axisymmetric processes will allow to simplify the mathematical processing of the obtained results. Real processes of metal processing are always accompanied by uneven plastic molding, which affects the power and kinematic parameters, the quality of finished products. The main technological factors affecting the appearance of the inhomogeneity and the nature of its distribution are the form factor (geometric parameters of the original workpiece) and the coefficient of contact friction. The study of technological parameters affecting the distribution of non-uniformity of plastic molding will allow to create practical recommendations for reducing the influence of the above mentioned parameters to ensure more uniform plastic deformation. Due to elastic-plastic deformation and local heating, which occur during surface-plastic deformation processing, the stress-strain and physical state of the surface layer is formed. This changes the microgeometry of the surface, the physical and mechanical properties of the surface layer of the workpiece. The surface treated by methods of surface-plastic deformation, has high hardness, residual compression stresses in the surface layer, smoothed out the micro-roughness. Accordingly, increases wear resistance, fatigue strength, resistance to corrosion and more.

Application of the parallel diagonal dominant algorithm for the incompressible Navier-Stokes equations

The accuracy and the applicability of the parallel diagonal dominant (PDD) algorithm are explored for highly scalable computation of the incompressible Navier-Stokes equations which are integrated using a fully-implicit fractional-step method in parallel computational environments. The PDD algorithm is known to be applicable only for an evenly diagonal dominant matrix. In the present study, however, it is shown mathematically that the PDD algorithm is utilizable even for non-diagonal dominant matrices derived from discretization of incompressible momentum equations. The order of accuracy and the error characteristics are investigated in detail in terms of the Courant-Friedrichs-Lewy (CFL) number and the grid spacing by conducting simulations of decaying vortices in both two and three dimensions, flow in a lid-driven cavity, and flow over a circular cylinder. In order to reduce communication cost, which is one of bottlenecks in parallel computation, an aggregative data communication method is combined with the PDD algorithm. Parallel performance of the present PDD-based method is investigated by measuring the speedup, efficiency, overhead, and serial fraction.

The interaction between the transient cavitating flow and hydrodynamic performance around a pitching hydrofoil

The interaction between the unsteady cavitating flow and hydrodynamic performance around a pitching Clark-Y hydrofoil is investigated experimentally and numerically. The experiments were conducted in the looped cavitation tunnel, and the cavitation patterns are documented by two high-speed digital cameras, and the moment of hydrofoil is measured by the moment sensor. The pitching hydrofoil is controlled from α+ = 10° to α+ = 15° firstly, and goes back to α− = 5° from α+ = 15°, then goes back to α+ = 10° from α− = 5° at Re = 4.4∗105. The upstream velocity U∞ and the cavitation number σ is fixed at 6.3 m/s and 1.38, respectively. And the pitching rate is α˙=40∘/s,α˙∗=0.086. The numerical investigations were performed by solving the incompressible UNRANS equations via the commercial code CFX using the Merkle cavitation model, the coupled k-ω SST turbulence model and γ-Reθ transition model. The predicted cavity patterns and moment coefficients agree well with the experimental results. The results showed there are two distinct cavitation patterns (Multi-scale cloud cavitation and Traveling sheet cavitation). For the Multi-scale cloud cavitation phase (α+ = 10°-α- = 10°), the re-entrant flow is the main factor on the stability of cavitating flow structures, which is responsible for different shedding patterns. According to the breaking position and re-entrant flow thickness, this stage is divided into three different patterns of the cavity development and shedding. For the Traveling sheet cavitation phase (α− = 10°-α+ = 10°), the shedding of cavity mainly results from the interaction of the re-entrant flow and the fluctuation of the gas liquid interface, thus leading to the irregular breaking points. The cavitating flow structure of different phases were further investigated using the LESs, which will be help to identify the dynamic behavior of flow structures effectively.