Incompressible Equations Research Articles

A Preliminary Study of Breaking Waves Phenomenon on a Numerical Wave Tank

The detailed prediction of breaking waves transformation may be a main concern within the ocean structures design and development at the top of submerged flat reefs. In this preliminary analysis, phenomenon of solitary wave on the reef can be simply interpreted in two-dimensional numerical wave tank (NWT), based on the computational fluid dynamics (CFD). A more sophisticated method, large eddy simulation (LES) turbulence model together with volume of fluid (VOF) for capturing free-surface, could clarify two-phase incompressible flow Navier-Stoke equations by finite volume approach. The other approaches have been used. Existing model results, wave generation, breaking wave, and velocity distributions, devised the validity of laboratory measurements. This set of tests desired to examine the effect of different reef configurations (reef length, reef height, incident wave height, and manning coefficient) on the solitary wave propagation. Results show that the model can contribute decent predictions of the phenomenon of breaking waves on various submerged flat reef configurations.

The Global Well-Posedness for Large Amplitude Smooth Solutions for 3D Incompressible Navier–Stokes and Euler Equations Based on a Class of Variant Spherical Coordinates

This paper investigates the globally dynamical stabilizing effects of the geometry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional (3D) incompressible Navier–Stokes (NS) and Euler systems. The global well-posedness for large amplitude smooth solutions to the Cauchy problem for 3D incompressible NS and Euler equations based on a class of variant spherical coordinates is obtained, where smooth initial data is not axi-symmetric with respect to any coordinate axis in Cartesian coordinate system. Furthermore, we establish the existence, uniqueness and exponentially decay rate in time of the global strong solution to the initial boundary value problem for 3D incompressible NS equations for a class of the smooth large initial data and a class of the special bounded domain described by variant spherical coordinates.

A fast pressure-correction method for incompressible flows over curved walls

We have developed an explicit and direct pressure-correction method for simulating incompressible flows over curved walls. In order to integrate the Navier Stokes (NS) equations in time, we have developed an explicit, three-stage, third-order Runge-Kutta based projection-method (FastRK3) which requires solving the Poisson equation for pressure only once per time step. We have chosen to discretize the incompressible NS equations written in the orthogonal coordinates rather than the general formulation in curvilinear coordinates because the former does not contain cross-derivatives in the advection, diffusion, Laplacian, and gradient operators. Thus, the computational cost of solving the NS equations is substantially reduced and the numerical stencils of the finite difference approximations to these operators mirror that of the Cartesian formulation. This property also allows us to develop an FFT-based Poisson solver for pressure (FastPoc) for those cases where the components of the metric tensor are independent of one spatial direction: surfaces of linear translation (e.g., curved ramps and bumps) and surfaces of revolution (e.g., axisymmetric ramps). We have verified and validated FastRK3 and we have applied FastRK3 for simulating separated flows over ramps and a bump. Finally, our results show that the new FFT-based Poisson solver, FastPoc, is thirty to sixty times faster than the multigrid-based linear solver (depending on the tolerance set for the multigrid solver), and the new flow solver, FastRK3, is overall four to seven times faster when using FastPoc rather than multigrid. In summary, given that the computational mesh satisfies the property of orthogonality, FastRK3 can simulate flows over curved walls with second-order accuracy in space.

Cavitating flow structures and corresponding hydrodynamics of a transient pitching hydrofoil in different cavitation regimes

The present paper applied experimental and numerical methods to investigate the cavitating flow structures and corresponding hydrodynamics for a transient pitching Clark-Y hydrofoil. The aims are to (1) improve the understanding of the interplay between the transient cavitating flow structures, motion of the hydrofoil, and hydrodynamic performance, (2) quantify the influence of cavitation on the hydrodynamic load coefficients and flow structures, and (3) analyze the evolution of cavitating flow during different cavitation regime. The pitching motion trajectory is a triangular wave with mean incidence of α0=10° and amplitude of Δα = 5° at a frequency of 2 Hz. The upstream velocity U∞ is fixed at 6.3 m/s, which is corresponding to Re=4.4 × 105. The cavitation patterns for different cavitation numbers are mainly documented by the high-speed photography, and the dynamic characteristics of the hydrofoil are measured by the torque sensor. The numerical investigations were performed by solving the incompressible URANS equations using the mass-transfer 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. Four typical regimes, including sub cavitation, inception cavitation, sheet cavitation and cloud cavitation, are observed. Compared to the sub cavitation case, the hydrodynamic coefficients and flow structures are significantly affected by the incipient cavity. Results show that the leading edge (LE) cavity promotes the formation of the counterclockwise tailing edge vortex (TEV), thus leading to decline of the lift. Moreover, the LE cavity also limits the formation of the clockwise second vortex (SV), which weakens the fluctuation of the hydrodynamic load. For the sheet cavitation case, three cavitating flow patterns(Pattern A/B/C) are observed in the hydrodynamic fluctuation stage, which is corresponding to different characteristic frequency. For the cloud cavitation case, the hydrodynamic curves present four distinct stages. According to the cavity breaking position and characteristic frequency, four different patterns(Pattern I/II/III/IV) of the cavity development and shedding are observed and analyzed.

Acceleration of high-order combined compact finite difference scheme for simulating three-dimensional flow and heat transfer problems in GPUs

In this article, the high-order upwinding combined compact difference scheme developed in a three-point grid stencil is applied to solve the incompressible Navier-Stokes (NS) and energy equations in three dimensions. The time integrator with symplectic property is employed to approximate the temporal derivative term in inviscid Euler equation so as to numerically retain the embedded Hamiltions and Casimir to get long-time accurate solutions. For the sake of computational efficiency in solving the three-dimensional NS equations, all the calculations will be accelerated using the hybrid CUDA and OpenAcc GPU programing models. The parallel speedup performance compared to the multicore of an Intel Xeon E5-2690V5 CPU is reported.

Free vibration of partially fluid-filled or fluid-surrounded composite shells using the dynamic stiffness method

In this paper, the dynamic stiffness method (DSM) is developed for the vibration analysis of laminated cylindrical and conical shells, which are partially filled or surrounded by quiescent fluid. The Reissner and Naghdi’s thin shell theory and incompressible fluid equations are employed on the structures and fluid, respectively, so as to characterize the fluid–structure coupling model. Shell elements with/without fluid loadings are derived and assembled by using the dynamic stiffness method. Numerical examples of the free vibration analysis for both empty and fluid-filled shells are validated against previous studies. Remarkable advantages regarding computational efficiency and accuracy of the DSM, especially in high-frequency domain, are presented through response curves compared with finite element calculations. As for partially fluid-filled or fluid-surrounded laminated shells, parametric studies are made to find the effects of material properties, fluid heights and semi-vertex angles on the dynamic characteristics.

On the well-posedness and decay characterization of solutions for incompressible electron inertial Hall-MHD equations

In this paper, by using the properties of decay character and the Fourier splitting method, we establish the time decay rates of solutions for three-dimensional incompressible electron inertial Hall-MHD equations. In particular, the optimal decay rates of the mixed space-time derivatives of the solution are obtained.

Analytical Estimation of Out-of-plane Strain in Ultrasound Elastography to Improve Axial and Lateral Displacement Fields.

Many types of cancers are associated with changes in tissue mechanical properties. This has led to the development of elastography as a clinically viable method where tissue mechanical properties are mapped and visualized for cancer detection and staging. In quasi-static ultrasound elastography, a mechanical stimulation is applied to the tissue using ultrasound probe. Using ultrasound radiofrequency (RF) data acquired before and after the stimulation, the tissue displacement field can be estimated. Elasticity image reconstruction algorithms use this displacement data to generate images of the tissue elasticity properties. The accuracy of the generated elasticity images depends highly on the accuracy of the tissue displacement estimation. Tissue incompressibility can be used as a constraint to improve the estimation of axial and, more importantly, the lateral displacements in 2D ultrasound elastography. Especially in clinical applications, this requires accurate estimation of the out-of-plane strain. Here, we propose a method for providing an accurate estimate of the out-of-plane strain which is incorporated in the incompressibility equation to improve the axial and lateral displacements estimation before elastography image reconstruction. The method was validated using in silico and tissue mimicking phantom studies, leading to significant improvement in the estimated displacement.

Solution of nearly incompressible field problems using a generalized finite element approach

It is well known that for problems approaching incompressible limits, standard finite element approaches suffer from suboptimal convergence due to a phenomenon known as locking. We analyze this phenomenon and present a robust generalized finite element formulation that enables optimal solution convergence. This approach is explored for the two-dimensional incompressible elasticity equations with a variable Poisson’s ratio, as well as the two-dimensional lid-driven cavity Stokes flow problem using a penalty pressure formulation.

A Regularity Criterion in Weak Spaces to Boussinesq Equations

In this paper, we study the regularity of weak solutions to the incompressible Boussinesq equations in R 3 × ( 0 , T ) . The main goal is to establish the regularity criterion in terms of one velocity component and the gradient of temperature in Lorentz spaces.

A mixed discontinuous Galerkin method for an unsteady incompressible Darcy equation

We study a mixed discontinuous Galerkin (MDG) method for solving a time-dependent Darcy problem, which simulates an incompressible fluid such as water flowing in a rigid porous medium. The discretization of the unsteady Darcy problem relies on a backward Euler scheme for temporal variable and MDG method for spatial variables. Spatially semi-discrete and fully discrete schemes are analyzed. Existence and uniqueness of the numerical solutions are proved, and optimal order error estimates are derived for both the velocity and pressure variables. Finally, some test problems are provided to display the performance of the MDG method, and numerical results are reported to support the theoretical predictions.

Global well-posedness to the Cauchy problem of 2D density-dependent micropolar equations with large initial data and vacuum

This paper studies the Cauchy problem of the 2D nonhomogeneous incompressible micropolar equations. It is shown that the problem admits a unique global strong solution. Note that the initial data can be arbitrary large and the initial density can contain vacuum states.

Global well-posedness of the 3D micropolar equations with partial viscosity and damping

In this paper, we are concerned with the Cauchy problem of the 3D incompressible micropolar equations with partial viscosity and damping. We establish the global existence and uniqueness of the solution with a general initial data.

Divergence-free radial kernel for surface Stokes equations based on the surface Helmholtz decomposition

In this paper, we develop and analyze a new divergence-free kernel approximation method for the time-dependent incompressible Stokes equations on surfaces. The novelty of our proposed method comes from the surface Helmholtz decomposition, which can convert the surface Stokes equations into a coupled equations in which the velocity is deadly divergence-free so that no inf–sup conditions have to be satisfied. Spatial discretization of velocity is implemented by the radial kernel divergence-free approximation spaces, which only need scatter nodes on surfaces. Using the radial basis function collocation method in space, we derive the rigorous stability and convergence result. Numerical examples are presented, demonstrating the efficiency of some model problems on more general surfaces.

Hydrodynamic characteristics and flow structures of pitching hydrofoil with special emphasis on the added force effect

The objective of this paper is to investigate the hydrodynamic characteristics and corresponding flow structures of a pitching Clark-Y hydrofoil with special emphasis on the added force effect. The experiments were performed in the looped cavitation tunnel, and the hydrodynamic characteristics are obtained by the dynamic moment measurement system. The average angle of attack and amplitude of pitching hydrofoil are 10° and 5°, respectively. The whole oscillatory motion is divided into two stages, namely the up stage and down stage. The pitching rate is set with the Reynolds number Re = 4.4 × 105. The incompressible URANS equations are solved by using the coupled k-ω SST turbulence model and γ-Reθ transition model. It can be shown that the numerical results agree well with the experimental measurements. Compared to the static hydrofoil, the lift of the pitching case is higher in the up stage because of the positive added lift. Meanwhile, the center of pressure is closer to the pitching axis. For the down stage, the lift of the pitching case is lower caused by the negative added lift and the pressure center is further away from the pitching axis. The main reason is that the different pitching direction corresponds to specific position of the added lift relative to the pitching axis, causing the pressure center to move in different directions. When the stall happens, the evolution of lift and the pressure center fluctuates due to the shedding vortex structures. Results show that the added force effect on the hydrodynamic force is negligible compared with the contributions from the vorticity within the flow in the stall phase. As the pitching rate increases, the added force effect becomes more significant, thus leading to the higher lift of the up stage and the lower lift of the down stage. Besides, for the stall phase, the dynamic stall angle is delayed for the fast pitching rate, which is due to the generation and development of the counterclockwise trailing edge vortex.

A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem

A novel notion for constructing a well-balanced scheme – a gradient-robust scheme – is introduced and a showcase application for the steady compressible, isothermal Stokes equations in a nearly-hydrostatic situation is presented. Gradient-robustness means that gradient fields in the momentum balance are well-balanced by the discrete pressure gradient — which is possible on arbitrary, unstructured grids. The scheme is asymptotic-preserving in the sense that it reduces for low Mach numbers to a recent inf–sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence of the coupled FEM–FVM scheme for the nonlinear, isothermal Stokes equations is proved by compactness arguments. Numerical examples illustrate the numerical analysis, and show that the novel approach can lead to a dramatically increased accuracy in nearly-hydrostatic low Mach number flows. Numerical examples also suggest that a straightforward extension to barotropic situations with nonlinear equations of state is feasible.

On the stability and bifurcation of the non-rotating Boussinesq equation with the Kolmogorov forcing at a low Péclet number

This study examines the stability and potential bifurcations of a stratified shear flow governed by the non-rotating incompressible Boussinesq equation at a low Péclet number. For the ratio of the vertical scale to the horizontal scale of a stratified flow a∈[3/4,3/2), it is shown that there exists a threshold for the Reynold number Re above which the steady stratified shear flow driven by the Kolmogorov forcing becomes linearly unstable. As a result, the Boussinesq equation exhibits a Hopf bifurcation. To further determine the type of the Hopf bifurcation, the model is reduced to a low-order system whose numerical analyses reveal that the Hopf bifurcation is supercritical. That is, a stable periodic solution emerges, which describes an oscillating thermal convection in a highly stratified shear flow arising in the atmosphere or interior of many stellar systems with low Péclet numbers.

Изотермические сдвиговые течения вязкой несжимаемой жидкости с пространственным ускорением при учете трех параметров Кориолиса

We study the solvability of the overdetermined system of Navier–Stokes equations, supple-mented by the incompressibility equation, which is used to describe isothermal large-scale shear flows of a rotating viscous incompressible fluid. Large–scale flows are studied in a thin-layer ap-proximation (the vertical velocity of the fluid is assumed to be zero). The rotation of a continuous fluid medium is described by three Coriolis parameters. The solution of the reduced system of Na-vier–Stokes equations is constructed in the Lin–Sidorov–Aristov class. In this case, both nonzero components of the velocity vector, the pressure and temperature fields are assumed to be full linear forms of two Cartesian coordinates, and the dependence on the third Cartesian coordinate has an arbitrary form (including non-polynomial). It is shown that the nonlinear overdetermined system of Navier–Stokes equations and of the incompressibility equation in the framework of the Lin–Sidorov–Aristov class reduces to the equivalent nonlinear overdetermined system of ordinary dif-ferential equations, in which the components of the hydrodynamic fields act as unknown functions. The compatibility condition for the equations of the resulting system is derived. It is shown that, if this compatibility condition is fulfilled, the system has a unique solution, and the spatial accelera-tions in both variables (the linearity with respect to them was postulated when choosing the solution class) prove to be constant functions. These results are a generalization of similar results obtained earlier in the study of solvability in the cases of one and two Coriolis parameters.

Global Regularity of the Three-Dimensional Fractional Micropolar Equations

The global well-posedness of the smooth solution to the three-dimensional (3D) incompressible micropolar equations is a difficult open problem. This paper focuses on the 3D incompressible micropolar equations with fractional dissipations $( \Delta)^{\alpha}u$ and $(-\Delta)^{\beta}w$.Our objective is to establish the global regularity of the fractional micropolar equations with the minimal amount of dissipations. We prove that, if $\alpha\geq \frac{5}{4}$, $\beta\geq 0$ and $\alpha+\beta\geq\frac{7}{4}$, the fractional 3D micropolar equations always possess a unique global classical solution for any sufficiently smooth data. In addition, we also obtain the global regularity of the 3D micropolar equations with the dissipations given by Fourier multipliers that are logarithmically weaker than the fractional Laplacian.

A novel solution method for unsteady incompressible Euler flow using the vorticity–Bernoulli-pressure formulation

A novel numerical formulation and time-integration method for the incompressible Euler flow equations is presented using the vorticity–Bernoulli-pressure system. Lagrangian advection of vorticity-carrying particles is adopted from classical vortex methods that are known to give accurate results for vortex-dominated flows and allow larger time steps than explicit Eulerian schemes. Solving only one scalar Poisson equation per time step instead of a vector-valued equation in vortex methods makes the new scheme theoretically more efficient on large computational grids. The equation of motion in rotation form is integrated in time under certain approximation assumptions and solved explicitly for the flow velocity. The velocity is eliminated as an unknown in the continuity equation, which is efficiently solved for the Bernoulli pressure using the predicted vorticity field obtained by Lagrangian advection. The Bernoulli pressure is efficiently obtained from the Poisson equation in finite volume formulation. An estimated velocity field is calculated from the equation of motion. To effectively counterbalance the discretization errors made by the prediction, in a corrector step the kinematic relation between vorticity and estimated velocity is reformulated and solved in a Lagrangian framework as well. The velocity field is corrected at the end of the time step. Two-dimensional cases on Cartesian grids are considered. Numerical results for three flow cases demonstrate the locally smooth and accurate behavior and the second-order spatial accuracy of the solution. The new scheme produces significantly less artificial vortex distortion than the classical vortex method. The favorable properties of the solutions with respect to accuracy and long-time stability of the overall kinetic energy and enstrophy are also emphasized.