Incompressible Equations Research Articles

Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions

Abstract We present in this paper a rigorous error analysis of the vector penalty-projection method for solving the time-dependent incompressible Stokes equations with open boundary conditions on part of the boundary. First, we prove the stability of the scheme. Then we provide an error analysis for the second-order vector penalty-projection method which shows that the convergence rate of the error on the velocity and the pressure is of order 2 in l ∞ ⁢ ( L 2 ⁢ ( Ω ) ) l^{\infty}(\mathbf{L}^{2}(\Omega)) and l 2 ⁢ ( L 2 ⁢ ( Ω ) ) l^{2}(L^{2}(\Omega)) respectively. In addition, it is shown that the splitting errors of the method varies as O ⁢ ( ε ) \mathcal{O}(\varepsilon) , where 𝜀 is a penalty parameter chosen as small as desired. Several numerical tests in agreement with the theoretical results are presented. To the best of our knowledge, this paper provides the first rigorous proof of optimal error estimates for second-order splitting schemes with open boundary conditions.

Least-squares pressure recovery in reduced order methods for incompressible flows

In this work, we introduce a method to recover the reduced pressure for Reduced Order Models (ROMs) of incompressible flows. The pressure is obtained as the least-squares minimum of the residual of the reduced velocity with respect to a dual norm. We prove that this procedure provides a unique solution whenever the full-order pair of velocity-pressure spaces is inf-sup stable. We also prove that the proposed method is equivalent to solving the reduced mixed problem with reduced velocity basis enriched with the supremizers of the reduced pressure gradients. Optimal error estimates for the reduced pressure are obtained for general incompressible flow equations and specifically, for the transient Navier-Stokes equations. We also perform some numerical tests for the flow past a cylinder and the lid-driven cavity flow which confirm the theoretical expectations, and show an improved convergence with respect to other pressure recovery methods.

A numerical model for solitary wave breaking based on the phase-field lattice Boltzmann method

This study presents a numerical investigation of a solitary wave breaking over a slope by using the phase-field lattice Boltzmann method. The incompressible two-phase flow equations are solved by using a velocity-based formulation of the two-phase lattice Boltzmann method with a central-moment collision model to accurately simulate wave breaking problems. For interface capture, a phase-field lattice Boltzmann method that ensures mass conservation is employed. The validity of the proposed method is confirmed through solitary wave propagation and transformation problems, and the obtained results are in good agreement with the experimental and calculated results. The proposed method is then employed to analyze wave breaking on a slope, demonstrating strong concordance with experimental data and existing computational findings. By analyzing the instantaneous flow characteristics and the temporal evolution of the variation in kinetic, potential, and total energy from deep to shallow water, the model can reveal the macroscopic characteristics of solitary wave breaking. Because the phase-field model effectively simulates wave breaking and air entrainment, it can depict wave energy dissipation more accurately than the single-phase lattice Boltzmann method with free surface tracking.

Lattice Boltzmann modeling of natural circulation loop with emphasis on non-Boussinesq mechanism

The natural circulation loop is crucial for the safe and stable operation of nuclear reactors and other applications. Traditional numerical algorithms, based on the Boussinesq approximation, have limitations when dealing with large temperature differences and density disparity, and they do not fully address fluid compressibility. This paper adopts the decoupled and stabilized lattice Boltzmann method (DSLBM) with a non-Boussinesq algorithm to study the natural circulation loop. The DSLBM provides a detailed flow description under large temperature and density differences, incorporating the pseudopotential multiphase model, temperature equation, and state equation, without relying on assumptions. The study examines the loop's performance under various temperature differences, central height differences, and heating source lengths, focusing on mass flow rate, driving head, and heating power. It reveals the energy performance, flow characteristics, and heat transfer properties of the loop, highlighting the physical mechanisms involved. Comparison with the empirical formulation of the incompressible equation from the theoretical aspect shows that when the temperature difference coefficient is lower than 0.15, the two methods are not much different from each other. When the temperature difference coefficient reaches 0.2, 0.3, and 0.4, the difference between the two methods is 9.47%, 19.11%, and 42.64%, respectively. Consequently, the Boussinesq approximation can be compensated by DSLBM, which proves the value of the application of the algorithm in exploiting the compressibility of fluids. The dimensionless fitting correlation with greater universality is obtained, which helps to predict the properties of the natural circulation loop with varying temperature differences, friction coefficients, and geometric structures. The research in this paper will lay the foundation for optimizing the system design of the natural circulation loop and improving energy utilization efficiency.

Simulations With Compressible Multiphase Formulation on Heat Transfer Studies in a Rocket Thrust Chamber With Experimental Validation

Abstract In the combustor for rocket engines, liquid film cooling is a widely adopted technique for restricting the structural components to the acceptable bounds for the successful operation of the propulsion systems. Multiple cooling techniques for thrust chamber walls are favored along with the coolant film in propulsion systems operating at higher chamber pressures by incorporating an ablative cooling mechanism for the nozzle. The adoption of combination will result in challenges in simulation studies due to the presence of multiphase phenomenon in the system. The current article presents the effects of these two cooling methods on a thrust chamber wall studied through compressible multiphase formulations. A majority of the previous studies have used incompressible flow equations to model coolant film behavior and associated heat transfer. The present study utilizes an approach utilizing compressible multiphase flow to simulate the behavior of the compressible hot gas flow and the incompressible coolant liquid film accounting for phase change effects, vapor diffusion into hot gas, radiation effects on coolant surface, liquid entrainment, and blowing effects due to vapor formation at interface. Film-cooling effects on ablative nozzle accounting for pyrolysis phenomenon due to material decomposition governed by Arrhenius expression are also emphasized. The outcome of the numerical model showed good agreement with available test data in literature, and results from the tests done in-house. The model described in the present study was able to support the actual evidence of liquid film cooling being able to preserve the walls of the thrust chamber from severe internal thermal environment and prevent ablation on surface of nozzle.

One-level and two-level operator splitting methods for the unsteady incompressible micropolar fluid equations with double diffusion convection

In this paper, a one-level operator splitting method (OOSM) and a two-level operator splitting method (TOSM) for the two-dimensional or three-dimensional (2D/3D) unsteady incompressible micropolar fluid equations with double diffusion convection (IMFDDC) are proposed and analyzed. Firstly, a OOSM is constructed, which consists of five steps. The projection strategy is adopted to get the values of linear velocity and pressure in the first two steps, then the three elliptic subproblems about the angular velocity, the temperature, and the concentration variables are solved in the last three steps. The original variables with the predefined boundary conditions need to be solved separately in each step. Next, in order to solve the problem efficiently, a TOSM is presented, which only solves the nonlinear equations in the coarse level subspaces and the Oseen type linearized equations in the fine level. The numerical analysis of stability and error estimates of the fully discrete methods show that TOSM can reach the same accuracy as the OOSM with H∼O(h2/3). Numerical examples are provided to confirm the effectiveness and reliability of the proposed methods.

The global existence for the strong solution of the stochastic 3-D incompressible anisotropic Naiver-Stokes equations

This paper considers the stochastic 3-D incompressible anisotropic Navier-Stokes equations. We establish the global existence of a strong solution in probability when either the horizontal component of the initial data and external force is sufficiently small or the horizontal viscosity coefficient is sufficiently large.

Optimized Schwarz waveform relaxation method for the incompressible Stokes problem

We propose and analyse the optimized Schwarz waveform relaxation (OSWR) method for the unsteady incompressible Stokes equations. Well-posedness of the local subdomain problems with Robin boundary conditions is proved. Convergence of the velocity is shown through energy estimates; however, pressure converges only up to constant values in the subdomains, and an astute correction technique is proposed to recover these constants from the velocity. The convergence factor of the OSWR algorithm is obtained through a Fourier analysis, and allows to efficiently optimize the space-time Robin transmission conditions involved in the OSWR method. Then, numerical illustrations for the two-dimensional unsteady incompressible Stokes system are presented to illustrate the performance of the OSWR algorithm.

A priori estimates of the unsteady incompressible thermomicropolar fluid equations and its numerical analysis based on penalty finite element method

In this paper, the unsteady incompressible thermomicropolar fluid (UITF) equations are considered. Theoretically, some a priori regularity conclusions are presented firstly, which seem to be not available in the literatures. Numerically, a penalty finite element method (PFEM) for the UITF equations is studied, the Euler semi-implicit temporal semi-discrete method of the penalty UITF equations is proposed, the stability and the L2-H1 error estimates of the temporal discrete solutions are proved. Finally, the stability and the L2-H1 error estimates for the finite element fully-discrete approximation of the penalty UITF equations are rigorously proved. The accuracy and efficiency of the fully-discrete PFEM are demonstrated by some numerical examples.

The two-and-a-half-dimensional incompressible Hall-MHD system with partial dissipation

In this paper, we investigate the Cauchy problem for the two-and-a-half-dimensional incompressible Hall-magnetohydrodynamic equations with partial dissipation. We prove the global regularity of a certain system of equations with horizontal or vertical dissipation and magnetic hyper-diffusion for large initial data.

Modelling and Impact Analysis of Combined Dam Failure of Small Tandem Reservoir Groups in Hilly Areas

Numerous small reservoirs in hilly areas play a significant role in flood control and profit-making in the basin, but there is also a risk of dam failure when encountering special circumstances such as excessive flooding, which may cause large losses downstream. In this paper, we take the small reservoirs in Shandong Province as the research object, analyze the hydraulic topology of three types of tandem reservoirs, generalize the shape of the breach by using the transient lateral local one-break to the end, calculate the dam-break flooding process by using the quadratic parabolic method, and simulate the dam-break by using the MIKE21 model based on the principle of the two-dimensional incompressible Reynolds mean stress equation for the incompressible fluid, and analyze the peak flow under various combinations of dam-break flow by setting various combinations of the number of dam-breaks and the timing of dam-break simulation. By setting various combinations of dam failure numbers and timing, we analyzed the changing rules of peak dam failure flow, inundation area and other elements under various combinations, and sought the most unfavorable combinations and parameters such as the timing of dam failure, and the results of the analyses can provide technical support for the assessment of the risk of dam failure in the tandem reservoirs in the hilly area and the risk correspondence.

Global well‐posedness of the 3D hydrostatic magnetohydrodynamics equations

In this paper, the three‐dimensional hydrostatic magnetohydrodynamic (HMHD) equations are considered on a thin domain. We showed the global existence and uniqueness (regularity) of strong solutions to the three‐dimensional incompressible HMHD equations without any small assumption on the initial data. More precisely, there exists a unique strong solution globally in time for any given ‐smooth initial data.

High order asymptotic inequalities for some dissipative systems

We obtain some important fundamental inequalities concerning the long time behavior of high order derivatives for solutions of some dissipative systems in terms of their algebraic L2 decay. Some of these inequalities have not been observed in the literature even for the fundamental Navier–Stokes equations. To illustrate this new approach, we derive bounds for the asymmetric incompressible fluids equations, where improved inequalities were established for the micro-rotational field w. Finally, we show how this technique can be applied to other dissipative systems.

Global well-posedness of three-dimensional incompressible Boussinesq system with temperature-dependent viscosity

In this paper, we focus on the global well-posedness of solutions to three-dimensional incompressible Boussinesq equations with temperature-dependent viscosity under the smallness assumption of initial velocity fields u0 in the critical space Ḃ3,10. The key ingredients here lie in the decomposition of the velocity fields and the regularity theory of the Stokes system, which are crucial to get rid of the smallness restricition of the initial temperature θ0. In addition, we mention that the improved decay estimates in time is also necessary.

A locally calculable P3‐pressure in a decoupled method for incompressible Stokes equations

AbstractThis article will suggest a new finite element method to find a ‐velocity and a ‐pressure solving incompressible Stokes equations at low cost. The method solves first the decoupled equation for a ‐velocity. Then, using the calculated velocity, a locally calculable ‐pressure will be defined component‐wisely. The resulting ‐pressure is analyzed to have the optimal order of convergence. Since the pressure is calculated by local computation only, the chief time cost of the new method is on solving the decoupled equation for the ‐velocity. Besides, the method overcomes the problem of singular vertices or corners.

Stability for a system of the 2D incompressible magneto-micropolar fluid equations with partial mixed dissipation

This paper focuses on the 2D incompressible anisotropic magneto-micropolar fluid equations with vertical dissipation, horizontal magnetic diffusion, and horizontal vortex viscosity. The goal is to investigate the stability of perturbations near a background magnetic field in the 2D magneto-micropolar fluid equations. Two main results are obtained. The first result is based on the linear system. Global existence for any large initial data and asymptotic linear stability are established. The second result explores stability for the nonlinear system. It is proven that if the initial data are sufficiently small, then the solution for some perturbations near a background magnetic field remains small. Additionally, the long-time behaviour of the solution is presented. The most challenging terms in the proof are the linear terms in the velocity equation and the micro-rotation equation that will grow with respect to time t. We are able to find some background fields to control the growth of the linear terms. Our results reveal that some background fields can stabilise electrically conducting fluids.

Structure and role of the pressure Hessian in regions of strong vorticity in turbulence

Amplification of velocity gradients, a key feature of turbulent flows, is affected by the non-local character of the incompressible fluid equations expressed by the second derivative (Hessian) of the pressure field. By analysing the structure of the flow in regions where the vorticity is the highest, we propose an approximate expression for the pressure Hessian in terms of the local vorticity, consistent with the existence of intense vortex tubes. Contrary to the often used simplification of an isotropic form for the pressure Hessian, which in effect inhibits vortex stretching, the proposed approximate form of the pressure Hessian enables much stronger vortex stretching. The prediction of the approximation proposed here is validated with results of direct numerical simulations of turbulent flows.

Investigation of the performance of a horizontal-axis dual rotor wind turbine

Recent years have seen a rise in interest in wind energy as a useful alternative to harmful energies like fossil fuels. The dual rotor wind turbine (DRWT) offers more rapid rates of wind energy extraction. The current study intends to compare the performance of the turbine with and without the addition of a second rotor. Additionally, it examines how tip speed ratio and phase shift angle will affect DRWT performance. Realizable k-shear stress transport turbulence models are used to solve the three-dimensional, turbulent, stable, and incompressible flow equations for the performance of dual-rotor wind turbines. Domain-independence tests and an impartial mesh test are run to assess the results and ensure their accuracy. The researcher relies on previous studies while constructing the single rotor wind turbine model. This model uses an S826 airfoil. The front and rear rotors are given streamlined representations using ANSYS, according to the researcher. The independent mesh test indicates that the mesh density has 11.5 million elements. The experiment's results show that the DRWT has a significant effect on the efficiency of wind energy.

Quantitative asymptotic stability of the quasi-linearly stratified densities in the IPM equation with the sharp decay rates

We analyze the asymptotic stability of the quasi-linearly stratified densities in the 2D inviscid incompressible porous medium equation on R2 with respect to the buoyancy frequency N. Our target density of stratification is the sum of the large background linear profile with its slope N and the small perturbation that could be both non-linear and non-monotone. Quantification in N will be performed not only on how large the initial density disturbance is allowed to be but also on how much the target densities can deviate from the purely linear density stratification without losing their stability.For the purely linear density stratification, our method robustly applies to the three fundamental domains R2, T2, and T×[−1,1], improving both the previous result by Elgindi (2017) [18] on R2 and T2, and the study by Castro et al. (2019) [12] on T×[−1,1]. The obtained temporal decay rates to the stratified density on R2 and to the newly found asymptotic density profiles on T2 and T×[−1,1] are all sharp, fully realizing the level of the linearized system. We require the initial disturbance to be small in Hm for any integer m≥4, which we even relax to any positive number m>3 via a suitable anisotropic commutator estimate.

Local well-posedness for incompressible neo-Hookean elastic equations in almost critical Sobolev spaces

Local well-posedness for incompressible neo-Hookean elastic equations in almost critical Sobolev spaces