Inflow-outflow Boundary Conditions Research Articles

Dissipative solutions to the model of a general compressible viscous fluid with the Coulomb friction law boundary condition

We study a model of a general compressible viscous fluid subject to the Coulomb friction law boundary condition. For this model, we introduce a dissipative formulation and prove the existence of dissipative solutions. The proof of this result consists of a three level approximation method: A Galerkin approximation, the classical parabolic regularization of the continuity equation as well as convex regularizations of the potential generating the viscouss stress and the boundary terms incorporating the Coulomb friction law into the dissipative formulation. This approach combines the techniques already known from the proof of the existence of dissipative solutions to a model of general compressible viscous fluids under inflow-outflow boundary conditions as well as the proof of the existence of weak solution to the incompressible Navier-Stokes equations under the Coulomb friction law boundary condition. It is the first time that this type of boundary condition is considered in the case of compressible flow.

No pressure? Energy-consistent ROMs for the incompressible Navier-Stokes equations with time-dependent boundary conditions

This work presents a novel reduced-order model (ROM) for the incompressible Navier-Stokes equations with time-dependent boundary conditions. This ROM is velocity-only, i.e. the simulation of the velocity does not require the computation of the pressure, and preserves the structure of the kinetic energy evolution.The key ingredient of the novel ROM is a decomposition of the velocity into a field with homogeneous boundary conditions and a lifting function that satisfies the mass equation with the prescribed inhomogeneous boundary conditions. This decomposition is inspired by the Helmholtz-Hodge decomposition and exhibits orthogonality of the two components. This orthogonality is crucial to preserve the structure of the kinetic energy evolution. To make the evaluation of the lifting function efficient, we propose a novel method that involves an explicit approximation of the boundary conditions with POD modes, while preserving the orthogonality of the velocity decomposition and thus the structure of the kinetic energy evolution.We show that the proposed velocity-only ROM is equivalent to a velocity-pressure ROM, i.e., a ROM that simulates both velocity and pressure. This equivalence can be generalized to other existing velocity-pressure ROMs and reveals valuable insights in their behaviour.Numerical experiments on test cases with inflow-outflow boundary conditions confirm the correctness and efficiency of the new ROM, and the equivalence with the velocity-pressure formulation.

Two-Phase Compressible/Incompressible Navier–Stokes System with Inflow-Outflow Boundary Conditions

We prove the existence of a weak solution to the compressible Navier–Stokes system with singular pressure that explodes when density achieves its congestion level. This is a quantity whose initial value evolves according to the transport equation. We then prove that the “stiff pressure” limit gives rise to the two-phase compressible/incompressible system with congestion constraint describing the free interface. We prescribe the velocity at the boundary and the value of density at the inflow part of the boundary of a general bounded \(C^2\) domain. For the positive velocity flux, there are no restrictions on the size of the boundary conditions, while for the zero flux, a certain smallness is required for the last limit passage.

Construction of weak solutions to compressible Navier–Stokes equations with general inflow/outflow boundary conditions via a numerical approximation

The construction of weak solutions to compressible Navier–Stokes equations via a numerical method (including a rigorous proof of the convergence) is in a short supply, and so far, available only for one sole numerical scheme suggested in Karper (Numer Math, 125(3):441–510, 2013) for the no slip boundary conditions and the isentropic pressure with adiabatic coefficient \(\gamma >3\). Here we consider the same problem for the general non zero inflow–outflow boundary conditions, which is definitely more appropriate setting from the point of view of applications, but which is essentially more involved as far as the existence of weak solutions is concerned. There is a few recent proofs of existence of weak solutions in this setting, but none of them is performed via a numerical method. The goal of this paper is to fill this gap. The existence of weak solutions on the continuous level requires several tools of functional and harmonic analysis and differential geometry whose numerical counterparts are not known. Our main strategy therefore consists in rewriting of the numerical scheme in its variational form modulo remainders and to apply and/or to adapt to the new variational formulation the tools developed in the theoretical analysis. In addition to the result, which is new, the synergy between numerical and theoretical analysis is the main originality of the present paper.

Steady-State Navier–Stokes Equations in Thin Tube Structure with the Bernoulli Pressure Inflow Boundary Conditions: Asymptotic Analysis

Steady-state Navier–Stokes equations in a thin tube structure with the Bernoulli pressure inflow–outflow boundary conditions and no-slip boundary conditions at the lateral boundary are considered. Applying the Leray–Schauder fixed point theorem, we prove the existence and uniqueness of a weak solution. An asymptotic approximation of a weak solution is constructed and justified by an error estimate.

Consistency, convergence and error estimates for a mixed finite element–finite volume scheme to compressible Navier–Stokes equations with general inflow/outflow boundary data

Abstract We study convergence of a mixed finite element-finite volume scheme for the compressible Navier–Stokes equations in the isentropic regime under the full range $1<\gamma <\infty $ of the adiabatic coefficients $\gamma $ for the problem with general nonzero inflow–outflow boundary conditions. We propose a modification of Karper’s scheme (2013, A convergent FEM-DG method for the compressible Navier–Stokes equations. Numer. Math., 125, 441–510) in order to accommodate the nonzero boundary data, prove existence of its solutions, establish the stability and uniform estimates, derive a convenient consistency formulation of the balance laws and use it to show the weak convergence of the numerical solutions to a dissipative solution with the Reynolds defect introduced in Abbatiello et al. (2021, Generalized solutions to models of compressible viscous fluids. Discrete Contin. Dyn. Syst., 41, 1--28). If the target system admits a strong solution then the convergence is strong towards the strong solution. Moreover, we establish the convergence rate of the strong convergence in terms of the size of the space discretization $h$ (which is supposed to be comparable with the time step $\varDelta t$). In the case of the nonzero inflow–outflow boundary data all results are new. The latter result is new also for the no-slip boundary conditions and adiabatic coefficients $\gamma $ less than the threshold $3/2$.

Assessing the Risk of Intracranial Aneurysm Rupture Using Morphological and Hemodynamic Biomarkers Evaluated from Magnetic Resonance Fluid Dynamics and Computational Fluid Dynamics.

Purpose:Evaluate in vivo hemodynamic and morphological biomarkers of intracranial aneurysms, using magnetic resonance fluid dynamics (MRFD) and MR-based patient specific computational fluid dynamics (CFD) in order to assess the risk of rupture.Methods:Forty-eight intracranial aneurysms (10 ruptured, 38 unruptured) were scrutinized for six morphological and 10 hemodynamic biomarkers. Morphological biomarkers were calculated based on 3D time-of-flight magnetic resonance angiography (3D TOF MRA) in MRFD analysis. Hemodynamic biomarkers were assessed using both MRFD and CFD analyses. MRFD was performed using 3D TOF MRA and 3D cine phase-contrast magnetic resonance imaging (3D cine PC MRI). CFD was performed utilizing patient specific inflow–outflow boundary conditions derived from 3D cine PC MRI. Univariate analysis was carried out to identify statistically significant biomarkers for aneurysm rupture and receiver operating characteristic (ROC) analysis was performed for the significant biomarkers. Binary logistic regression was performed to identify independent predictive biomarkers.Results:Morphological biomarker analysis revealed that aneurysm size [P = 0.021], volume [P = 0.035] and size ratio [P = 0.039] were statistically significantly different between the two groups. In hemodynamic biomarker analysis, MRFD results indicated that ruptured aneurysms had higher oscillatory shear index (OSI) [OSI.max, P = 0.037] and higher relative residence time (RRT) [RRT.ave, P = 0.035] compared with unruptured aneurysms. Correspondingly CFD analysis demonstrated significant differences for both average and maximum OSI [OSI.ave, P = 0.008; OSI.max, P = 0.01] and maximum RRT [RRT.max, P = 0.045]. ROC analysis revealed AUC values greater than 0.7 for all significant biomarkers. Aneurysm volume [AUC, 0.718; 95% CI, 0.491–0.946] and average OSI obtained from CFD [AUC, 0.774; 95% CI, 0.586–0.961] were retained in the respective logistic regression models.Conclusion:Both morphological and hemodynamic biomarkers have significant influence on intracranial aneurysm rupture. Aneurysm size, volume, size ratio, OSI and RRT could be potential biomarkers to assess aneurysm rupture risk.

Compressible Navier–Stokes system with hard sphere pressure law and general inflow–outflow boundary conditions

We prove the existence of a weak solution to the compressible Navier–Stokes system with hard sphere possibly non-monotone pressure law involving, in particular, the Carnahan–Starling model [2] largely employed in various physical and industrial applications. We take into account large velocities prescribed at the boundary of a bounded piecewise C2 domain and large densities prescribed at the inflow boundary without any restriction neither on the shape of the inflow/outflow boundaries nor on the shape of the domain.

The structure and propagation of laminar flames under autoignitive conditions

The laminar flame speed sl is an important reference quantity for characterising and modelling combustion. Experimental measurements of laminar flame speed require the residence time of the fuel/air mixture (τf) to be shorter than the autoignition delay time (τ). This presents a considerable challenge for conditions where autoignition occurs rapidly, such as in compression ignition engines. As a result, experimental measurements in typical compression ignition engine conditions do not exist. Simulations of freely propagating premixed flames, where the burning velocity is found as an eigenvalue of the solution, are also not well posed in such conditions, since the mixture ahead of the flame can autoignite, leading to the so called “cold boundary problem”. Here, a numerical method for estimating a reference flame speed, sR, is proposed that is valid for laminar flame propagation at autoignitive conditions. Two isomer fuels are considered to test this method: ethanol, which in the considered conditions is a single-stage ignition fuel; and dimethyl ether, which has a temperature-dependent single- or two-stage ignition and a negative temperature coefficient regime for τ. Calculations are performed for the flame position in a one-dimensional computational domain with inflow-outflow boundary conditions, as a function of the inlet velocity UI and for stoichiometric fuel–air premixtures. The response of the flame position, LF, to UI shows distinct stabilisation regimes. For single-stage ignition fuels, at low UI the flame speed exceeds UI and the flame becomes attached to the inlet. Above a critical UI value, the flame detaches from the inlet and Lf becomes extremely sensitive to UI until, for sufficiently high UI, the sensitivity decreases and Lf corresponds to the location expected from a purely autoignition stabilised flame. The transition from the attached to the autoignition regimes has a corresponding peak dLf/dUI value which is proposed to be a unique reference flame speed sR for single-stage ignition fuels. For two-stage ignition fuels, there is an additional stable regime where a high-temperature flame propagates into a pool of combustion intermediates generated by the first stage of autoignition. This results in two peaks in dLf/dUI and therefore two reference flame speed values. The lower value corresponds to the definition of sR for single-stage ignition fuels, while the higher value exists only for two-stage ignition fuels and corresponds to a high temperature flame propagating into the first stage of autoignition and is denoted sR′. A transport budget analysis for low- and high-temperature radical species is also performed, which confirms that the flame structures at UI=sR and UI=sR′ do indeed correspond to premixed flames (deflagrations), as opposed to spontaneous ignition fronts which do not have a unique propagation speed.

An existence result for a system of coupled semilinear diffusion-reaction equations with flux boundary conditions

In this paper, we consider diffusion, reaction and dissolution of mobile and immobile chemical species present in a porous medium. Inflow–outflow boundary conditions are considered at the outer boundary and the reactions amongst the species are assumed to be reversible which yield highly nonlinear reaction rate terms. The dissolution of immobile species takes place on the surfaces of the solid parts. Modelling of these processes leads to a system of coupled semilinear partial differential equations together with a system of ordinary differential equations (ODEs) with multi-valued right-hand sides. We prove the global existence of a unique positive weak solution of this model using a regularization technique, Schaefer's fixed point theorem and Lyapunov type arguments.

Inflow–outflow boundary conditions along arbitrary directions in Cartesian lake models

Specifying point sources and sinks of water near boundaries is presented as a flexible approach to prescribe inflows and outflows along arbitrary directions in Cartesian grid lake models. Implementing the approach involves a straightforward modification of the governing equations, to include a first order source term in the continuity and momentum equations. The approach is implemented in a Cartesian grid model and applied to several test cases. First, the flow along a straight flat bottom channel with its axis forming different angles with the grid directions is simulated and the results are compared against well-known analytical solutions. Point-sources are then used to simulate unconfined inflows into a reservoir (a small river entering a reservoir in a jet-like manner), which occur at an angle with the grid directions. The model results are assessed in terms of a mixing ratio between lake and river water, evaluated at a cross section downstream of the inflow boundary. Those results are particularly sensitive to changes in the inflow angle. It is argued that differences in mixing rates near the inflow sections could affect the fate of river-borne substances in model simulations.

A weakly compressible MPS method for modeling of open‐boundary free‐surface flow

AbstractA mesh‐free particle method, based on the moving particle semi‐implicit (MPS) interaction model, has been developed for the simulation of two‐dimensional open‐boundary free‐surface flows. The incompressibility model in the original MPS has been replaced with a weakly incompressible model. The effect of this replacement on the efficiency and accuracy of the model has been investigated. The new inflow–outflow boundary conditions along with the particle recycling strategy proposed in this study extend the application of the model to open‐boundary problems. The final model is able to simulate open‐boundary free surface flow in cases of large deformation and fragmentation of free surface. The models and proposed algorithms have been validated and applied to sample problems. The results confirm the model's efficiency and accuracy. Copyright © 2009 John Wiley & Sons, Ltd.

Flows of weakly compressible viscoelastic fluids through a regular bounded domain with inflow-outflow boundary conditions

We study steady isothermal motions of a nonlinear weakly compressible viscoelastic fluids of Oldroyd type ina bounded domain $\Omega\subset\mathbb{R}^2$, with given non-zerovelocities on the boundary of $\Omega$. We suppose that the pressure$p$ and the extra-stress tensor $\tau$ are prescribed on the part ofthe boundary corresponding to entering velocities. A uniqueness andexistence result for the solution $(\mathbf u,p,\tau)$ isestablished in $W^{2,q}(\Omega)\times W^{1,q}(\Omega)\timesW^{1,q}(\Omega)$ with $ 2 < q < 3$. The proof follows from an analysis ofa linearized problem. The fixed point theorem is used to establishthe existence of a solution. The solutions of two transportequations for $p$ and $\tau$ are obtained by integration along thestreamlines.

Mixing and reacting flow with inflow–outflow boundary conditions

We consider the behaviour of mixing reacting compressible flows with inflow–outflow boundary conditions corresponding to the injection of reactants, fuel and oxidizer in a bounded region. Analytical results on the existence of solutions for small time and data are given in the two-dimensional case, using extensions of the techniques of Valli and Zajaczowski [Navier–Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case, Commun. Math. Phys. 103 (1986), 259–296]. As well, computational results are presented using finite difference methods.

Solving hyperbolic conservation laws using multiquadric quasi-interpolation

In this article, we apply the univariate multiquadric (MQ) quasi-interpolation to solve the hyperbolic conservation laws. At first we construct the MQ quasi-interpolation corresponding to periodic and inflow-outflow boundary conditions respectively. Next we obtain the numerical schemes to solve the partial differential equations, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the differential equation and a low-order explicit difference to approximate the temporal derivative of the differential equation. Then we verify our scheme for the one-dimensional Burgers' equation (without viscosity). We can see that the numerical results are very close to the exact solution and the computational accuracy of the scheme is 𝒪(τ), where τ is the temporal step. We can improve the accuracy by using the high-order quasi-interpolation. Moreover the methods can be generalized to the other equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

Compressible Navier–Stokes Model with Inflow-Outflow Boundary Conditions

In the paper [7], author gives a definition of weak solution to the nonsteady Navier–Stokes system of equations which describes compressible and isentropic flows in some bounded region Ω with influx of fluid through a part of the boundary ∂Ω. Here, we present a way for proving existence of such solutions in the same situation as in [7] under the sole hypothesis γ > 3/2 for the adiabatic constant.

Combining Cauchy and characteristic codes. V. Cauchy-characteristic matching for a spherical spacetime containing a perfect fluid

This paper is part of a long term program to develop CCM (combined Cauchy and characteristic) codes as investigative tools in numerical relativity. The approach has two distinct features: (i) it dispenses with an outer boundary condition and replaces this with matching conditions at an interface between the Cauchy and characteristic regions, and (ii) by employing a compactified coordinate, it proves possible to generate global solutions. In this paper it is shown that CCM can be used effectively to model a spherically symmetric perfect fluid. A particular advantage of CCM in avoiding arbitrary mass inflow-outflow boundary conditions is pointed out. Results are presented which include fluid distributions which form black holes and those which give rise to mass outflow.

Stream-function based multiple Bluff Bodies 2D flow analysis

The structure of 2D solenoidal or non solenoidal incompressible viscous channel flows in presence of obstacles was numerically investigated with special reference to laminar and transition conditions. The results obtained with a ψ,γ model and a Control Volume approach depend upon the supposed inflow-outflow boundary conditions. The numerical solution of the flow around multiple Bluff Bodies require also the determination of the B.C.s for the Stream Function on the obstacles. Moreover, for models of incompressible flows, where also source like obstacles are considered, the flow may result to be non solenoidal; in these cases multivalued Stream Functions were successfully employed. A new, highly accurate, vector upwinded scheme, recently introduced by the authors, is used for the treatment of convective terms at high Cell Reynolds numbers. Meaningful results were obtained without the need of special grids, or of spectral methods.

Approximate nonreflecting inflow-outflow boundary conditions for the calculation of navier-stokes equations in internal flows

In this paper, the nonreflecting boundary conditions based upon fundamental ideas of the linear analysis are developed for gas dynamic equations, and the modified boundary conditions for Navier-Stokes equations are proposed as a substitute of the nonreflecting boundary conditions inside boundary layers near rigid walls. These derived boundary conditions are then applied to calculations both for the Euler equations and the Navier-Stokes equations to determine if they can produce acceptable results for the subsonic flows in channels. The numerical results obtained by an implicit second-order upwind difference scheme show the effectiveness and generality of the boundary conditions. Furthermore, the formulae and the analysis performed here may be extended to three dimensional problems.

Inflow-outflow boundary conditions for two-dimensional acoustic waves in channels with flow

An analysis of the number and form of the required inflow-outflow boundary conditions for the full two-dimensional time-dependent nonlinear acoustic system in subsonic mean flow is performed. The analysis is performed in a rectangular channel with rigid walls and is extendable to more complicated geometries and wall conditions. The explicit predictor-corrector method of MacCormack is used to integrate the acoustic system coupled with the derived boundary conditions. The methodology is tested on both uniform and sheared mean flows with plane and nonplanar sources